Math Focused Questions

Which of the following quadratic equation has roots − (2/3) and − (3/5)?

Recall that the quadratic equation can be expressed as𝑥𝑥2 (𝑠𝑠𝑠𝑠𝑠𝑠 𝑜𝑜𝑜𝑜 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟) +(𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑜𝑜𝑜𝑜 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟) = 0. From the given information, sum of the roots = − 23 −35 = −10−915 =− 1915. Product of the roots = − 23× − 35 = 25 . Therefore, 𝑥𝑥2 − − 1915 𝑥𝑥 + 25 = 0 ⇒ 15𝑥𝑥2 + 19𝑥𝑥 +6 = 0

Solve for 𝑚𝑚: 10𝑚𝑚 − 8 − 4𝑚𝑚 + 2 = 16𝑚𝑚 + 4

Solving for 𝑚𝑚 in the equation: 10𝑚 − 8 − 4𝑚 + 2 = 16𝑚 + 4, rearrange and combine terms on each side to get 6𝑚 − 6 = 16𝑚 + 4. Subtracting 4 and 6𝑚 from both the sides, gives −10 = 10𝑚 ⇒ 𝑚 = −1 (𝐵).

Which of the following equations illustrates the Commutative Law of Addition?

The commutative law of addition states that 𝑋 + 𝑌 = 𝑌 + 𝑋. Option A uses the commutative law of multiplication, option B the additive inverse law, option D the distributive law, and option E is just an addition. Only 𝐶, 157 + 4 = 4 + 157, illustrates the commutative law of addition.

Find the degree of the polynomials 3x^7 + 9x^6 × x^2 − 5x^4 × x + 19x^3 − 12x^2 + 25x + 7 = 0

The degree of the polynomial is the highest power of the polynomial. The polynomial 3𝑥𝑥7 + 9𝑥𝑥6 × 𝑥𝑥2 − 5𝑥𝑥4 × 𝑥𝑥 + 19𝑥𝑥3 − 12𝑥𝑥2 + 25𝑥𝑥 + 7 = 0 can be re-written as 9𝑥𝑥8 + 3𝑥𝑥7 − 5𝑥𝑥5 + 19𝑥𝑥3 − 12𝑥𝑥2 + 25𝑥𝑥 + 7 = 0, in which the highest power is 8. The degree of the polynomial is 8,

Determine the value of 𝑝𝑝 for which the equation 4x^2 + 6 + 2p = 0 has a real root.

The quadratic equation has real roots, which means 𝑏𝑏2 − 4𝑎𝑎𝑎𝑎 ≥ 0. From the given equation: 𝑎𝑎 = 4, 𝑏𝑏 = 6 𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐 = 2𝑝𝑝, hence 62 − 4 × 4 × 2𝑝𝑝 ≥ 0 ⇒ 36 − 32𝑝𝑝 ≥ 0 ⇒ 36 ≥ 32𝑝𝑝. Therefore, 𝑝𝑝 ≤ 98

Subtract (−4𝑥^2 + 6𝑥 − 18) from (8𝑥^2 − 12𝑥 − 18).

The resulting equation is (8𝑥𝑥2 − 12𝑥𝑥 − 18) − (−4𝑥𝑥2 + 6𝑥𝑥 − 18). Grouping like terms gives(8 + 4)𝑥𝑥2 − (12 + 6)𝑥𝑥 − 18 + 18 ⇒ 12𝑥𝑥2 − 18

Find (5x^2 + 4x^3 + 14x^7 + 14) − (4x^3 + 5 − 13x^2).

To find (5𝑥𝑥2 + 4𝑥𝑥3 + 14𝑥𝑥7 + 14) − (4𝑥𝑥3 + 5 − 13𝑥𝑥2), group like terms and simplify.⇒ 14𝑥𝑥7 + (4 − 4)𝑥𝑥3 + (5 + 13)𝑥𝑥2 + (14 − 5) ⇒ 14𝑥𝑥7+18𝑥𝑥2 + 9,

Fill the missing term: (8x^2 + 42x + 16) + (3x^2 − [? ]x + 6) = 11x^2 + 20x + 22

To find the missing term, we just need to equate the coefficients of all terms with 𝑥𝑥 on both sides of the equation, which gives 42𝑥𝑥 − [? ]𝑥𝑥 = 20. Hence the missing term is 22

If a+b=35, b+c=15 and a+c=25, then which of the following is true?

To find which is greater than the other, we simply subtract equations. (𝑎 + 𝑏) − (𝑏 + 𝑐) = 𝑎 − 𝑐 = 20 𝑎 > 𝑐 (𝑎 + 𝑐𝑐) − (𝑏 + 𝑐) = 𝑎 − 𝑏 = 10 ;𝑎 >b (𝑎 + 𝑏) − (𝑎 + 𝑐) = 𝑏 − 𝑐 = 10 ; 𝑏 > 𝑐 Using the three results above, we get 𝑎 > 𝑏 > 𝑐, which is option (𝐸).

Find the product of (2 − 𝑥 + 𝑥^2), (2 − 𝑥^2 + 𝑥^3) and (1 + 𝑥 + 𝑥^2)

(2 − 𝑥𝑥 + 𝑥𝑥2)(2 − 𝑥𝑥2 + 𝑥𝑥3) (1 + 𝑥𝑥 + 𝑥𝑥2) = [(2)(2 − 𝑥𝑥2 + 𝑥𝑥3) − (𝑥𝑥)(2 − 𝑥𝑥2 + 𝑥𝑥3) + (𝑥𝑥2)(2 − 𝑥𝑥2 + 𝑥𝑥3)](1 + 𝑥𝑥 + 𝑥𝑥2) =(4 − 2𝑥𝑥2 + 2𝑥𝑥3 − 2𝑥𝑥 + 𝑥𝑥3 − 𝑥𝑥4 + 2𝑥𝑥2 − 𝑥𝑥4 + 𝑥𝑥5)(1 + 𝑥𝑥 + 𝑥𝑥2) ⇒ (𝑥𝑥5 − 2𝑥𝑥4 + 3𝑥𝑥3 − 2𝑥𝑥 + 4)(1 + 𝑥𝑥 + 𝑥𝑥2) ⇒ (𝑥𝑥5)(1 + 𝑥𝑥 + 𝑥𝑥2) − (2𝑥𝑥4)(1 + 𝑥𝑥 + 𝑥𝑥2) + (3𝑥𝑥3)(1 + 𝑥𝑥 + 𝑥𝑥2) − (2𝑥𝑥)(1 + 𝑥𝑥 + 𝑥𝑥2) + (4)(1 + 𝑥𝑥 + 𝑥𝑥2) ⇒ 𝑥𝑥5 + 𝑥𝑥6 + 𝑥𝑥7 − 2𝑥𝑥4 − 2𝑥𝑥5 − 2𝑥𝑥6 + 3𝑥𝑥3 + 3𝑥𝑥4 + 3𝑥𝑥5 − 2𝑥𝑥 − 2𝑥𝑥2 − 2𝑥𝑥3 + 4 + 4𝑥𝑥 + 4𝑥𝑥2 ⇒ 𝑥𝑥7 − 𝑥𝑥6 + 2𝑥𝑥5 + 𝑥𝑥4 + 𝑥𝑥3 + 2𝑥𝑥2 + 2𝑥𝑥 + 4,

Expand (8x^2 − 8x − 14)(x + 3)

(8𝑥𝑥2 − 8𝑥𝑥 − 14)(𝑥𝑥 + 3) = (𝑥𝑥)(8𝑥𝑥2 − 8𝑥𝑥 − 14) + (3)(8𝑥𝑥2 − 8𝑥𝑥 − 14) = 8𝑥𝑥3 − 8𝑥𝑥2 − 14𝑥𝑥 +24𝑥𝑥2 − 24𝑥𝑥 − 42 = 8𝑥𝑥3 + 16𝑥𝑥2 − 38𝑥𝑥 − 42,

Find the missing term: (𝑥 + 1)(𝑥^4 + 3𝑥^3 + 4𝑥^2 − 8) = 𝑥^5 + 4𝑥^4 + [? ] + 4𝑥^2 − 8𝑥 − 8 .

(𝑥𝑥 + 1)(𝑥𝑥4 + 3𝑥𝑥3 + 4𝑥𝑥2 − 8) = (𝑥𝑥)(𝑥𝑥4 + 3𝑥𝑥3 + 4𝑥𝑥2 − 8) + (1)(𝑥𝑥4 + 3𝑥𝑥3 + 4𝑥𝑥2 − 8) = 𝑥𝑥5 + 3𝑥𝑥4 + 4𝑥𝑥3 − 8𝑥𝑥 + 𝑥𝑥4 + 3𝑥𝑥3 + 4𝑥𝑥2 − 8 = 𝑥𝑥5 + 4𝑥𝑥4 + 7𝑥𝑥3 + 4𝑥𝑥2 − 8𝑥𝑥 − 8. Comparing this expression to the right hand side shows that [?] = 7𝑥𝑥3

Which of the following expressions is a cubic polynomial?

A cubic polynomial is a polynomial of degree 3, i.e. the highest power is 3

Which of the following laws of real numbers does (z+12)+(-z-12)=0 illustrate?

An expression plus its negative gives the identity element (0). Since the two brackets in the given statement cancel each other out, it expresses the additive inverse law that any real number 𝑥 has - 𝑥 such that the two sum to 0 , option 𝐶.

Find the remainder when 𝑝(𝑥) = 4𝑥^3 − 12𝑥^2 + 14𝑥 − 3 is divided by 𝑔(𝑥) = 𝑥 − (1/2)

By remainder theorem, we know that 𝑝𝑝(𝑥𝑥) when divided by 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 − 12, gives a remainder equal to 𝑝𝑝 12. 𝑝𝑝(𝑥𝑥) = 4𝑥𝑥3 − 12𝑥𝑥2 + 14𝑥𝑥 − 3 ⇒ 𝑝𝑝 12 = 4 123− 12 122+ 14 12− 3 = 48 −124+ 142 − 3 = 12 − 3 + 7 − 3 = 32

Two numbers sum to 60 and their ratio is 2/3 . Find the larger of the two numbers.

D Let one of the numbers be 𝑥. The sum of both the numbers is 60. Therefore the second number is 60 − 𝑥. Their ratio is given as (2/3). You can set up the equation 𝑥/( 60−𝑥) = (2/ 3). Cross-multiplying gives 3𝑥 = 120 − 2𝑥 ⇒ 5𝑥 = 120 ⇒ 𝑥 = 24, 60 − 𝑥 = 36. The larger number is 36, option D.

FInd the greatest positive integer that will divide 398, 436 and 542 leaving remainders 7, 11 and 15 respectively.

Dividing 398 by the required number gives a remainder of 7. This means 398 − 7 = 391 is exactly divisible by the number, i.e. it is a factor of 391. Similarly, the integer is a factor of 436 − 11 = 425 and 542 − 15 = 527. The required number is the HCF (Highest Common Factor) of 391, 425 and 527. 391 = 17 × 23, 425 = 25 × 17 and 527 = 17 × 31. 17 is the common factor, option C.

Which of the following terms is a factor of x^2 + 2x − 48 = 0?

Factorize the given equation 𝑥^2 + 2𝑥 − 48 = 0 by splitting the middle term as 𝑥^2 + 8𝑥 − 6𝑥 − 48 = 0. This equation can be expressed as (𝑥 + 8) − 6(𝑥 + 8) = 0 ⇒ (𝑥 + 8)(𝑥 − 6) = 0. Only the first term matches an answer option

What must be added to the polynomial 𝑓(𝑥) = 𝑥^5 + 𝑥^4 + 3𝑥^3 − 6𝑥^2 − 4𝑥 + 8 so that the resulting polynomial is exactly divisible by 𝑔(𝑥) = 𝑥 − 2

From the division algorithm 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥)𝑞𝑞(𝑥𝑥) + 𝑟𝑟(𝑥𝑥), 𝑓𝑓(𝑥𝑥) + {−𝑟𝑟(𝑥𝑥)} = 𝑔𝑔(𝑥𝑥)𝑞𝑞(𝑥𝑥) to obtain a polynomial divisible by 𝑔𝑔(𝑥𝑥). By long division, 5 4 418---------------------- - 4x 8 4x +10---------------------- 4x 8 4x 4 - 5x 10- x 2xx 3xx 2x2 x x 3x 6 4x +104 3 223 23 24 34 35 45 4 3 2+ + + +−−−− − − − − − − − − − − − − − −− −+− − − − − − − − − − − − − − − −− − − + − −x x x xxxxxx xThe remainder is 𝑟𝑟(𝑥𝑥) = 18. Therefore, we need to add −𝑟𝑟(𝑥𝑥) = −18 to 𝑓𝑓(𝑥𝑥)

The polynomials 𝑓(𝑥) = 𝑎𝑥3 − 7𝑥^2 + 7𝑥 − 2 and 𝑔(𝑥) = 𝑥^3 − 2𝑎𝑥^2 + 8𝑥 − 8 when divided by 𝑥 − 2 leave the same remainder. Find the value of 𝑎.

From the division algorithm, 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥)𝑞𝑞(𝑥𝑥) + 𝑟𝑟(𝑥𝑥) ⇒ 𝑓𝑓(𝑥𝑥) − 𝑟𝑟(𝑥𝑥) = 𝑔𝑔(𝑥𝑥)𝑞𝑞(𝑥𝑥). We need to subtract the remainder of the division of 𝑓𝑓(𝑥𝑥) by 𝑔𝑔(𝑥𝑥). Use long division: x xx26x______________- ( 4x 2x)4x 8x______________2x x2 1 2x 3x 8x2223 23 2−−− −− −− ++ − −𝑟𝑟(𝑥𝑥) = −6𝑥𝑥, option �

Which of the following expressions is equal to: (7x + 3y) − (3𝑦 + 15) + (−7x + 12)?

Group like terms and simplify. (7𝑥𝑥 + 3𝑦𝑦) − (3𝑦𝑦 + 15) + (−7𝑥𝑥 + 12) ⇒ 7𝑥𝑥 − 7𝑥𝑥 + 3𝑦𝑦 − 3𝑦𝑦 − 15 + 12 ⇒ 0 + 0 − 3 ⇒ −3,

Simplify ((5/4)a + 7b) + (6.4b − 2a) − ((7/8)a − 15b).

Group like terms and simplify. 54𝑎𝑎 + 7𝑏𝑏 + (6.4𝑏𝑏 − 2𝑎𝑎) − 78𝑎𝑎 − 15𝑏𝑏 ⇒ 54 − 2 − 78 𝑎𝑎 +(7 + 6.4 + 15)𝑏𝑏 ⇒ −1.625𝑎𝑎 + 28.4𝑏𝑏,

Find the sum of the polynomials (2x^3 + 5x + 5x^2 + 4) and (6x^3 + 5x^2 + 16).

Group the like terms and simplify . (2𝑥𝑥3 + 5𝑥𝑥 + 5𝑥𝑥2 + 4) + (6𝑥𝑥3 + 5𝑥𝑥2 + 16) ⇒ (2 + 6)𝑥𝑥3 + (5 + 5)𝑥𝑥2 + 5𝑥𝑥 + 20 ⇒ 8𝑥𝑥3 + 10𝑥𝑥2 + 5𝑥𝑥 + 20

Which of the following numbers is divisible by both 2 or 3?

Here we simply check all the answer choices for a number that is divisible by both 2 and 3. Options 𝐴, 𝐵,𝐷 and 𝐸 are divisible by 2 and options 𝐶 and 𝐸 are divisible by 3. Only 𝐸, 1122 is divisible both by 2 and 3.

The linear equation that converts Fahrenheit to Celsius is F= (9/5)C+32 . If the temperature is 95 𝐹, the temperature in Celsius is

If F = 95, 95 = (9/5)𝐶 + 32 ⇒ 63 = (9/5)𝐶 ⇒ 𝐶 = 35 (option A)

Which of the following laws of real numbers does 8(x+7)=8x+56 illustrates?

In the statement 8(𝑥 + 7) = 8𝑥 + 56 , the right side is obtained by taking the elements of addition inside the bracket and multiplying each by the element in front of the bracket. This action follows the distributive law that 𝑥(𝑦 + 𝑧) = 𝑥y + 𝑥z, option 𝐴.

Which of the following numbers are an irrational number?

Irrational numbers cannot be expressed as a ratio of two numbers. The given answer options can be written as √2√2 = 2, √676 = 24, 0.75 = (3/4), √1331 3 = 11. 7√5 cannot be reduced to a fraction: it gives an infinitely repeating decimal..

Adam sold his Macbook and accessories for $400. If he received seven times as much money for the Macbook as he did for the accessories, how much did he receive for the Macbook?

Let Adam receive $𝑥 for selling his accessories. He receives 7 times more money for his Macbook, i.e. $7𝑥. The total amount he receives is $400. Therefore, 7𝑥 + 𝑥 = 400 ⇒ 8𝑥 = 400 ⇒ 𝑥 = 50. $7𝑥 = 7 × 50 = $350 (𝐸).

20 years from now Jim's age will be 5 times his present age. Jim's present age is

Let Jim's present age be 𝑥𝑥. After 20 years his age will be 5𝑥. Therefore, 𝑥 + 20 = 5𝑥 ⇒ 20 = 4𝑥 ⇒ 𝑥 = 5 (B).

You have 10l of a solution at 25% concentration and want to obtain a 30% concentration. How much of a solution at 40% concentration should you add?

Let the amount of 40% solution required to be mixed be (𝑥𝑥)𝑙𝑙, implying there will be (𝑥𝑥 + 10)𝑙𝑙 of 30% solution after mixing the two. The required equation is (0.4)(𝑉 of 40% 𝑠) + (0.25)(𝑉 of 25% s) = (0.3)(V of 30% ). Plugging in the values gives 0.4𝑥 + 0.25(10) = 0.3(𝑥 + 10) ⇒ 0.4𝑥 + 2.5 = 0.3𝑥 + 3 ⇒ 0.1𝑥 = 0.5 ⇒ 𝑥 = 5𝑙. Hence, we require 5𝑙𝑙 of the 40% solution to obtain a 30% solution, which is option 𝐸.

The unequal side of an isosceles triangle is 5cm longer than its equal sides. If the perimeter of the triangle is 20 cm, find the length of the unequal side.

Let the equal sides of the isosceles triangle be 𝑥 cm long. The unequal side is 5𝑐m more i.e. 𝑥 + 5 cm. The perimeter is the sum of all three sides. Therefore, 𝑥 + 𝑥 + 𝑥 + 5 = 20, ⇒ 3𝑥 = 15, 𝑥= 5. 𝑥 + 5 = 10 cm, option A.

The sum of a number and its reciprocal is 10/3 . Find the number

Let the number be 𝑥𝑥 such that its reciprocal is 1𝑥𝑥. Set up the equation 𝑥𝑥 + 1𝑥𝑥 = 103 sidsides both sidesothsides by 3𝑥𝑥 ⇒ (3𝑥𝑥2 + 3) = 10𝑥𝑥 ⇒ 3𝑥𝑥2 − 10𝑥𝑥 + 3 = 0. Split the middle term ⇒ 3𝑥𝑥2 − 9𝑥𝑥 − 𝑥𝑥 +3 = 0 ⇒ 3𝑥𝑥(𝑥𝑥 − 3) − 1(𝑥𝑥 − 3) = 0 ⇒ (3𝑥𝑥 − 1)(𝑥𝑥 − 3) = 0. Hence, 𝑥𝑥 = 3𝑜𝑜𝑜𝑜 𝑥𝑥 = 13

In a classroom the number of boys exceeds the number of girls by 20. If the total number of students is 80, find the number of boys in the class.

Let the number of girls and boys in the classroom be 𝑥𝑥 and 𝑥 + 20, respectively. Since there are 80 students total, 𝑥 + 𝑥 + 20 = 80 ⇒ 2𝑥 = 80 − 20 = 60. 𝑥 = 30, and the number of boys in the class = 30 + 20 = 50, option

The sum of the ages of a father and his son is 50 years. Five years ago, the product of their ages was 175. How old is the son now?

Let the present age of father be 𝑥𝑥 years. Then the present age of son is (50 − 𝑥𝑥) years. Five years ago the father was (𝑥𝑥 − 5) years old and the son was 45 − 𝑥𝑥 years old. The product of their ages then was 175. of their ages is 175. Therefore, (𝑥𝑥 − 5)(45 − 𝑥𝑥) = 175 ⇒ 45𝑥𝑥 − 225 − 𝑥𝑥2 + 5𝑥𝑥 = 175 ⇒ 50𝑥𝑥 − 225 − 𝑥𝑥2 = 175 ⇒ −𝑥𝑥2 + 50𝑥𝑥 − 400 = 0. This equation factors as −(𝑥𝑥 − 40)(𝑥𝑥 − 10) = 0, 𝑥𝑥 = 40 or 𝑥𝑥 = 10

Let 𝑃(𝑥) = (𝑥^3 + 𝑥^2 − 4𝑥 − 4) and 𝐾(𝑥) = (𝑥 − 2). Find the quotient when 𝑃(𝑥) is divided by 𝐾(𝑥) with no remainder.

Let 𝑃𝑃(𝑥𝑥) = 𝐾𝐾(𝑥𝑥)𝑞𝑞(𝑥𝑥) + 𝑟𝑟(𝑥𝑥). When 𝑃𝑃(𝑥𝑥) = (𝑥𝑥3 + 𝑥𝑥2 − 4𝑥𝑥 − 4) is divided by 𝐾𝐾(𝑥𝑥) = (𝑥𝑥 − 2) with no remainder, (𝑥𝑥3 + 𝑥𝑥2 − 4𝑥𝑥 − 4) = (𝑥𝑥 − 2)𝑞𝑞(𝑥𝑥) + 0 ⇒ 𝑞𝑞(𝑥𝑥) = 𝑥𝑥3+𝑥𝑥2−4𝑥𝑥−4 (𝑥𝑥−2) . Via long division we get ⇒ 𝑞𝑞(𝑥𝑥) = 𝑥𝑥2 + 3𝑥𝑥 + 2

Convert x=1.272727 ... to fractional form.

Let 𝑥 = 1.27272727 .... As there are two repeating digits after the decimal point, multiply both sides by 100 to give 100𝑥 = 127.272727. Subtracting the first equation gives 99𝑥 = 126, 𝑥 = 14 11, option 𝐵.

Solve for x: 4(5x-2)=16x.

Multiply out the left side of the equation: 4(5𝑥 − 2) = 16𝑥 ⇒ 20𝑥 − 8 = 16𝑥. Adding 8 to both the sides gives 20𝑥 = 16𝑥 + 8. Next, subtract 16𝑥𝑥 from both sides ⇒ 4𝑥 = 8,𝑥 = 2, which is option

Find the root(s) of the equation 2x^2 − 12x + 32 = 4x.

Rearrange and combine the terms of the equation to get 2(𝑥𝑥2 − 8𝑥𝑥 + 16) = 0. This equation factors as (𝑥𝑥 − 4)2 and has only one real root

Find the discriminant of the quadratic equation 2x^2 + 7x = 4.

Recall that the discriminant = 𝑏𝑏2 − 4𝑎𝑎𝑎𝑎. In the given equation, 𝑎𝑎 = 2, 𝑏𝑏 = 7 𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐 = −4. Substitute the values into the former equation to get 49 − 4(2)(−4) = 49 + 32 = 81

Which of following terms is a solution of the equation x^2 − (11/4)x + (15/8) = 0?

Solving for 𝑥𝑥 in the equation 𝑥𝑥2 − 114𝑥𝑥 + 158 = 0, split the middle term as − 54𝑥𝑥 − 64𝑥𝑥. 𝑥𝑥2 − 54𝑥𝑥 − 32𝑥𝑥 + 158 = 0 ⇒ 𝑥𝑥 𝑥𝑥 − 54 − 32 𝑥𝑥 − 54 = 0 ⇒ 𝑥𝑥 − 32𝑥𝑥 − 55 = 0 ⇒ 𝑥𝑥 = 32𝑜𝑜𝑜𝑜 𝑥𝑥 = 54

Find all possible values of x if x^2 - 3x = 10.

Start by subtracting 10 from both the sides, giving 𝑥^2 − 3𝑥 − 10 = 0. Factorize the equation by splitting the middle term ⇒ 𝑥^2 − 5𝑥 + 2𝑥 − 10 = 0 ⇒ 𝑥(𝑥 − 5) + 2(𝑥 − 5) = 0 ⇒ (𝑥 + 2)(𝑥 −5) = 0. 𝑥 = −2 𝑥 = 5

Find the coefficient of x^2 in 4x^3 + x^2 − 4c = 0.

The coefficient of 𝑥𝑥2 in 4𝑥𝑥3 + 𝑥𝑥2 − 4𝑐𝑐 = 0 is simply the absolute value before the unknown, which in this case is (1)𝑥𝑥2

Find two irrational numbers between sqrt(2) and sqrt(7).`

The squares of √2 and √7 are 2 and 7 respectively. As 2 < 3 < 5 < 7, it follows that √2 < √3 <√5 < √7. Therefore √3 𝑎𝑎𝑎𝑎𝑎𝑎 √5 , both irrational numbers, lie between √2 and√7 - option 𝐴. Note that the square root of 6.25 lies in this range but is a rational number.

Expand (3𝑥^3 + 6𝑥^2 + 12)(6𝑥^3 + 3𝑥 + 3).

To expand (3𝑥𝑥3 + 6𝑥𝑥2 + 12)(6𝑥𝑥3 + 3𝑥𝑥 + 3), distribute the first bracket's terms: (3𝑥𝑥3)(6𝑥𝑥3 +3𝑥𝑥 + 3) + (6𝑥𝑥2)(6𝑥𝑥3 + 3𝑥𝑥 + 3) + (12)(6𝑥𝑥3 + 3𝑥𝑥 + 3). Expand each bracket to obtain 18𝑥𝑥6 +9𝑥𝑥4 + 9𝑥𝑥3 + 36𝑥𝑥5 + 18𝑥𝑥3 + 18𝑥𝑥2 + 72𝑥𝑥3 + 36𝑥𝑥 + 36. Rearranging and combining like terms gives 18𝑥𝑥6 + 36𝑥𝑥5 + 9𝑥𝑥4 + 99𝑥𝑥3 + 18𝑥𝑥2 + 36𝑥𝑥 + 36,

Solve for y:4(y+5)=6(y-8).

To solve for 𝑦𝑦 in 4(𝑦 + 5) = 6(𝑦 − 8), multiply out both brackets to get 𝑦 + 20 = 6𝑦 − 48. Add 48 to both sides, giving 4𝑦𝑦 + 68 = 6𝑦𝑦. Subtract 4𝑦 from both sides ⇒ 68 = 2𝑦, 𝑦 = 34, which is option 𝐶𝐶.

Find 𝑚 if 8 is a root of the equation 2x^2 − 10x − m = 0.

We substitute 8 for 𝑥𝑥 in the equation 2𝑥^2 − 10𝑥 − 𝑚 = 0 and get 2(8)^2 − 10(8) − 𝑚 = 0 ⇒128 − 80 − 𝑚 = 0 ⇒ 48 − 𝑚 = 0.𝑚 = 48

𝑃(𝑥) = (𝑥^3 + 3𝑥^2 − 𝑘x + 4) divided by 𝐾(𝑥) = 𝑥 − 2, gives the remainder 𝑘. Find k

𝑃𝑃(𝑥𝑥) = (𝑥𝑥3 + 3𝑥𝑥2 − 𝑘𝑘𝑘𝑘 + 4). By remainder theorem, when 𝑓𝑓(𝑥𝑥) is divided by (𝑥𝑥 − 2), the remainder = 𝑓𝑓(2). 𝑓𝑓(2) = 23 + 3 × 22 − 2𝑘𝑘 + 4 = 8 + 12 − 2𝑘𝑘 + 4 = 24 − 2𝑘𝑘. Setting the remainder equal to this expression, 24 − 2𝑘𝑘 = 𝑘𝑘 ⇒ 3𝑘𝑘 = 24 ⇒ 𝑘𝑘 = 8

If ab=5, bc=(2/11) and ac=9, what is one possible value of abc?

𝑎b × 𝑏c × 𝑎c gives 𝑎^2𝑏^2𝑐^2 = 8.18. Taking the square root gives us abc = ±2.86 (D).

Find the value of the polynomial 𝑓(x) = 49x^3 + 18x^2 + 42x + 16 when x = 3

𝑓𝑓(𝑥𝑥) = 49𝑥𝑥3 + 18𝑥𝑥2 + 42𝑥𝑥 + 16. Substituting the value 𝑥𝑥 = 3 into the equation gives (49 × 33) + (18 × 32) + (42 × 3) + 16 = 745,

What must be subtracted from 𝑓(𝑥) = 2𝑥^3 − 3𝑥^2 − 8𝑥 to make it exactly divisible by 𝑔(𝑥) = 2𝑥 + 1?

𝑥𝑥3 − 6𝑥𝑥 = 0 ⇒ 𝑥𝑥(𝑥𝑥2 − 6) = 0 ⇒ 𝑥𝑥�𝑥𝑥 + √6��𝑥𝑥 − √6� = 0. ∴ 𝑥𝑥 = 0, ±√6 (𝐵𝐵).