ACT study guide Math: Polynomial Operations and Factoring Simple Quadratic Equations

Simplify the following expression: 2a^5b^6c/6a^3b^2

(2a^5b^6c/6a^3b^2=(1/3)a(5−3)b(6−2)c=)a2b4c/3

(1/3a - b)^2 =

To solve this problem, use the FOIL (First, Outside, Inside, Last) method for binomial multiplication to expand (1/3a−b)^2. It may be helpful to picture (1/3a−b)2 as (1/3a−b)(1/3a−b). Using the FOIL method, (1/3a−b)(1/3a−b) becomes (1/9)a^2−(1/3)ab−(1/3)ab+b^2=(1/9)a^2−(2/3)ab+b^2

Which of the following is equivalent to the expression 1−x^2/x for all nonzero values of x?

1−x^2/x = 1/x − x^2/x = 1/x − x

(3x2 − 5x + 1) − (3x2 − 2x + 6) =

((3x2 − 5x + 1) − (3x2 −2x + 6) = 3x2 − 5x + 1 − 3x2 + 2x − 6 =) − 3x − 5

The expression (x − a)(x + a)

(Using the FOIL method, (x − a) (x + a) = x2 + ax − ax − a2 =) x2 - a2

What are the values for a that satisfy the equation (a+y)(a+z)=0?

(In order for the expression (a+y)(a+z) to equal zero, either a+y=0 or a+z=0. Thus either) a=−y or a=−z

Which of the following is equivalent to the expression 3a2 − 2b2 + b?

(Since the terms are not like terms, they cannot be combined at all, and answer choices A and B can be eliminated. In answer choices C and D, the term −b has been factored out of 2b2 + b, and only answer choice D does so correctly. Answer choice E tries to factor out 2b, but 2 is not a factor of b.) 3a2 − b(2b − 1)

For what values of x is x2 − 3x − 10 = 0?

(The quadratic function x2 − 3x − 10 factors into (x − 5)(x + 2). This can equal zero only if) x = 5 or x = −2

(x3 + 2x2 + 3x − 2) − (2x3 −x2 − 4) is equivalent to

(To solve this problem, distribute the −1 and add like terms. After distributing, the expression (x3 + 2x2 + 3x − 2) − (2x3 − x2 − 4) becomes x3 + 2x2 + 3x − 2 − 2x3 +x2 + 4. Combining like terms yields) −x3 + 3x2 + 3x + 2

A number x can be written in the form k2 − 2 for some integer value of k. Which of the following expressions represents x2 in terms of k?

To find the square of x, you must use the FOIL technique: x2 = (k2 − 2)2 = k4 - 4k2 + 4

For all values of a and b, which of the following expressions is equivalent to (a − 6) (b + 4)?

(Using the FOIL method, (a − 6)(b + 4) =) ab + 4a − 6b − 24

If x = 4 and y = −1, then xy4 + x3y =

(xy4 +x3y = (4)( −1)4 + 43(−1) = 4 − 64 =) −60

For x≠0, 2x/9 × 3/x =

(2x/9 × 3/x = 6x/9x =)23

If one factor of a polynomial function is x − k, then which of the following must be a zero of the function?

(By the zero product rule, if the function is (x − k)(f(x)), then x − k = 0 and f(x) = 0 can be used to find the zeros of the function. Using the first equation, x = k must be a zero of the function.) K

If 2x2 − 6x = −4 and x < 0, then x =

(Dividing both sides of the equation by 2 gives an equivalent equation, x2 − 3x = −2, which can be restated as x2 − 3x + 2 = 0. Factor the equation to get (x − 2)(x − 1) = 0. By the zero product rule, x = 2 and x = 1. Therefore, there is no value x < 0 that makes the equation true.) There is no such value of x

What is the largest value of x that makes the equation x2 + 2x − 35 = 0 true?

(Factoring gives x2 + 2x − 35 = (x + 7) (x − 5) = 0. By the zero product rule, x + 7 = 0 and x − 5 = 0. Therefore, the roots are −7 and 5. The question asks for the largest value, which is) 5.

If x > 0 and 2x2 − 5x + 3 = x2 − 5x + 2, then x =

(The equation 2x2 − 5x + 3 = x2 − 5x + 2 is equivalent to the equation x2 + 1 = 0. This equation is equivalent to x2 = −1, which has) no real solutions.

Which of the following is a factor of x4 − 16?

(The expression x4 − 16 is a difference of squares, so it factors into (x2 + 4)(x2 − 4), which factors further into (x2 + 4)(x + 2)(x − 2).)x − 2

What is the greatest common factor of 2a4 and a3?

(The greatest common factor can be thought of as the "largest" term that will divide both terms. In this case, a3 is the largest term that can divide both terms.) a3

How many roots larger than 5 does x2 − 3x + 2 have?

(The roots of an expression are the solutions to that expression set equal to zero. Since this expression factors, the roots can be found by solving (x − 2)(x − 1) = 0. Of the roots 2 and 1, neither is larger than 5.) none

When x≠0,x2−4x^2+x^2/x=

(The terms in the numerator all have the same variable and the same degree, so they are like terms and can be added across: x2−4x2+x2x=−2x2x. Since the numerator and the denominator share a factor of x, this is equivalent to )−2x

4x3 × 3xy2 × 2xy2 is equivalent to

(To find an equivalent expression, Multiply the constants (4 × 3 × 2 = 24) Combine the x terms ( x3 × x × x ) → x3+1+1 → x5 Combine the y terms (y2 × y2 → y2+2 → y4). The result is) 24x5 y4

5x3 × 2xy × 3xy2 is equivalent to

(To find the equivalent expression: Multiply the constants (5 × 2 × 3). Combine the x terms (x3)(x)(x) = x3 + 1 + 1 = x5 (because when you have a common base you keep the base and add the exponents). Combine the y terms (y1)(y2) = y1 + 2, or y3. The result is) 30x5y3

Which of the following is the product of (3x2 − 1) (x2 − 4)?

(To solve this problem, distribute using the FOIL method, as follows: (3x2−1)(x2−4) First: (3x2)(x2) = 3x4 Outside: (3x2)(−4) = −12x2 Inside: (−1)(x2) = −x2 Last: (−1)(−4) = 4 Combine like terms and simplify to get) 3x4 − 13x2 + 4

If r and s are constants and x2 + rx + 12 is equivalent to (x + 3) (x + s), what is the value of r?

(To solve this problem, multiply the expression (x + 3) by (x + s) to get x2 +3x + sx + 3s. You are given that x2 +rx + 12 is equivalent to x2 + 3x + sx + 3s. Therefore, 3s is equal to 12, making s equal to 4. It is also apparent that 3x + sx is equivalent to rx. Set the quantities equal and solve for r, as follows: rx=3x+sxrx=x(3 +s)r=3+s Because s = 4, r must equal) 7

For all n, (3n + 5)2 = ?

(To solve, use the FOIL (First, Outside, Inside, Last) method for binomial multiplication to expand (3n + 5)2. It may be helpful to picture (3n + 5)2 as (3n + 5)(3n + 5). Using the FOIL method, (3n + 5)(3n + 5) becomes 9n2 + 15n + 15n + 25, which can be reduced to )9n2 + 30n + 25

In the (x, y) coordinate plane, which of the following functions would have a graph that crosses or touches the x-axis at only one point?

(To touch the x-axis at only one point and be a quadratic function as shown in the answer choices, the function would have to be of the form (x + c)2 = x2 + 2xc + c2 or (x − c)2 = x2 − 2xc + c2 for some value of c. The function in answer choice C is of this form, since 8 = 2(4) and 42 = 16.) x2 − 8x + 16

If k = −1, then (x+y/k)^2=

(When k=−1,(x+y/k)2=(−1)^2(x+y)2=(x+y)2 . By using the FOIL technique this expression can be simplified to) x2 + 2xy + y2

When x/y=4, x^2 −12y^2 =?

(You are given that x/y=4, so x = 4y. Substitute 4y for x in the equation and solve, as follows: x2 − 12y2 = (4y)2 − 12y2 = 16y2 − 12y2 )= 4y2