ACT study guide Math: Roots of Polynomials - Factoring to Find Solutions

How many real number solutions does the equation 2x3 = 9 have?

(The given equation is equivalent to x^3=9/2, which has one solution, x=3^√9/2) 1

If 3x2−12x=0 and x > 0, what is the value of x?

(We can factor x out of the equation 3x2−1/2x=0 to get the equivalent equation x(3x−1/2)=0. Since this is the product of two terms, we can then apply the zero product rule to find x = 0 or 3x−1/2=0. The second equation has a solution of )x=1/6

Which of the following is a root of −5x + x2?

(A root of a function is a solution to the equation formed when setting the function equal to zero. In this case, it would be a solution to −5x + x2 = 0, which is equivalent to x(−5 + x) = 0 when x is factored out. Finally, by the zero product rule, the roots are) x = 0 and x = 5

For what positive value of x is 3x2 − 19x = 14?

(After rewriting the equation as 3x2 − 19x − 14 = 0, we see that it factors into (3x + 2) and (x − 7). Therefore, the only positive solution will be x =) 7

For fixed real values of a, b, and c, a solution to the equation ax2 + bx + c = 0 is −5/2. For the same values of a, b, and c, which of the following is a factor of the expression ax2 + bx + c?

(At some point in solving the given equation, the equation x=−52 must have been found. Work backward by adding 52 to both sides and multiplying both sides by 2: x+52=0, and 2x + 5 =) 0

For a certain quadratic equation, ax2 + bx + c = 0, the 2 solutions are x=3/4 and x=−2/5. Which of the following could be factors of ax2 + bx + c?

(For a certain quadratic equation ax2 + bx + c = 0, if x=ab is a solution, then a possible factor would be (bx − a). Since two solutions for ax2 + bx + c = 0 are x=34 and x=−25, then possible factors are) 4x − 3 and 5x + 2

In the standard (x, y) coordinate plane, a polynomial function f(x) crosses the x-axis at the points (−3, 0), (−2, 0), (4, 0), (5, 0), and (7, 0) only. Each of the following is a factor of f(x) EXCEPT

(If f(x) is a polynomial with the given zeros, then (x + 3), (x + 2), (x − 4), (x − 5), and (x − 7) are factors of f(x)) x − 3

The value of a square's area is twice as large as the value of its perimeter. In units, what is the length of one side of the square?

(If the length of any one side of the square is x, then the area is x2, and the perimeter is 4x. If the area is twice the perimeter, x2 = 2(4x), and x2 − 8x= 0. By factoring, we see this equation is equivalent to x(x − 8) = 0, which has solutions 0 and 8. The length cannot be 0, so it must be )8 units

The number representing the length of one side of a square is 20% as large as the number representing its area. What is the perimeter of this square?

(Let A be the area of the square and s be the length of one side. We are told that s = 0.2A; since this is a square, A = s2 and s = 0.2s2. Bringing all the terms to one side and factoring, we have s(0.2s − 1) = 0. The solutions are s = 0 and s=10.2=5. The sides can't have a length of zero, so each side has a length of 5, and the perimeter is 4 × 5 =) 20

Which of the following expressions is a prime polynomial factor of a16 − 16?

(Remember that a difference of squares factors easily, such as: a2 − b2 = (a + b)(a − b). Using the same technique, you can factor a16 − 16 into (a8 + 4)(a8 − 4). The factor (a8 − 4) is another difference of squares, so it can be factored further into itself: (a8 − 4) = (a4 + 2)(a4 − 2). Of these factors, only (a4 + 2) is an answer choice.) a4 + 2

If the function f(x) = x2 − 9x + 8 is graphed in the (x, y) coordinate plane, then for which of the following values of x will the graph of f(x) be below the x-axis?

(Since the coefficient on the first term is positive, the graph of the function is a parabola opening up. Additionally, since x2 − 9x + 8 = (x − 8)(x − 1), the function crosses the x-axis at x = 8 and x = 1. Therefore, for values of x between 1 and 8, f(x) will be negative.) 1 < x < 8

For all m and n, (3m + n)(m2− n) = ?

(Since this problem requires you to multiply two binomials, you can utilize the FOIL (First, Outside, Inside, Last) method to multiply the expressions. First:(3m)(m2)=3m3Outside:(3m)(−n)=−3mnInside:(n)(m2)=m2nLast:(n)(−n)=−n2 Finally, add all these terms up to come up with your final answer. (3m + n)(m2 − n) =) 3m3 − 3mn + m2n − n2

Which of the following points is on the graph of the function f(x) = (x − 5)(x + 3)(x − 1)(x + 10)?

(Since x − 5 is a factor, 5 must be a zero of the function, and (5, 0) is on its graph.) 5, 0

What is the product of the 2 solutions of the equation x2 + 4x − 21 = 0?

(The easiest way to solve this problem is to remember that when two binomial expressions are multiplied, there is a predictable result. (x + a)(x − b) = x2 − bx + ax − ab. (x + a)(x − b) = 0, then x = −a and x = b. The product of the solutions is −ab. With this expression, x2 + 4x − 21 = 0: (x + 7)(x − 3) = 0 x = −7, x = 3 The product of the solutions (−3 × 7) is )−21

Which of the following is a value of r for which (r + 2)(r − 3) = 0?

(The expression (r + 2)(r − 3) will equal zero when either r + 2 or r − 3 equals zero. Thus r = −2 or r = 3.) -2

Which of the following functions has the same roots as f(x) = 3x2 − 3x − 27?

(The function f(x) = 3x2 − 3x − 27 = 0 is equivalent to 1/3(3x^2−3x−27)=1/3(0), or x^2−x−9=0) f5(x) = x2 − x − 9

What are the zeros of the following function? f(x) = x2 − 5x − 14

(The zeros are the values of x that make f (x) = zero. f(x) = x2 − 5x − 14 0 = (x − 7)(x + 2)) x = 7, x = − 2

Which of the following is a solution to the equation x2 + 25x = 0?

(To solve the quadratic equation x2 + 25x = 0 for x, factor out an x on the left side of the equation: x(x + 25). Now, apply the zero product rule: x = 0 or x + 25 = 0. If x + 25 = 0, then x = )−25

A rectangle has sides of lengths x and x + 1, where x is a positive number. If the area of the rectangle is 12, then which of the following is equivalent to the ratio of x to x + 1?

(Using the area of the rectangle, we can write x(x + 1) = 12, or x2 + x − 12 = 0. Factoring, we see that (x + 4)(x − 3) = 0, and the equation has the solutions x = −4 and x = 3. Since the length must be positive, x = 3, and the ratio of x to x + 1 is 3 to 4.) 3:4

Which of the following is a zero of the function f(x) = x2 (x − 4) (x + 1)?

(Using the zero product rule, the zeros of the function are the solutions to x2 = 0, x − 4 = 0, and x + 1 = 0. These solutions are 0, 4, and −1.) 0

What is one value of x that satisfies this equation? (x^3)2/3 −2x=3

(x3)23=x3×23=x2 x2−2x=3 x2−2x−3=0 (x−3)(x+1)=0 x=3,x=−1

If f(x) = x3, which of the following tables would correctly represent the values of −f(x) when x is −2, −1, 0, 1, and 2?

Given that −f(x) = −x3, then... For x = −2, we find −f(−2) = −(−2)3 = −(−8) = 8. For x = −1, we find −f(−1) = −(−1)3 = −(−1) = 1. For x = 0, the solution is −f(0) = −(0)3 = 0. For x = 1, it is −f(1) = −(1)3 = −(1) = −1, For x = 2, it is −f(2) = −(2)3 = −8

(Since the function is a polynomial that does not cross the x-axis at any other point, it will be of the form (x + a)(x + b)(x − c) for positive a, b, and c, since the x-coordinates of A and B are negative while the x-coordinate of C is positive. Only choice A meets this condition.) f(x) = (x + 3) (x + 4) (x − 1)

The graph of a polynomial function crosses the x-axis at the points A, B, and C as indicated in the following figure. If the graph does not cross the x-axis at any other point, which of the following could be the formula for the function?

Which of the following expressions represents a polynomial with exactly one real root?

x2 + 2x + 1 = (x + 1)2 ((x + 1)2 = 0 square root both sides: x + 1 = 0 subtract 1: x = −1 which has a single root of x = −1)

For all values of x < 4, the function f(x)=x^2−3x−4/x−4 has the same value as

x^2−3x−4/x−4=(x−4)(x+1)/x−4=x+1,x≠4