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Factoring - Answers
1. D
2. B
3. D
4. C
5. A
6. A
7. B
8. B
9. A
10. C
11. C
12. A
13. D
14. B
15. B
Factoring - Explanations
1.
The polynomial x2 - 7x + 12 is in the form ax2 + bx + c, where a = 1,
b = -7, and c = 12. Find the factors of a and c.
1: 1
12: 1, 2, 3, 4, 6, 12
Form two pairs of factors from one factor of a and one factor of c, so that when the factors are
multiplied and then added together (because c is positive), they equal the absolute value of b.
(1 · 4) + (1 · 3) = 7
Use these numbers to form the factors of the polynomial. Because b is negative, both factors will
have subtraction signs.
(x - 4)(x - 3)
Therefore, (x - 4)(x - 3) is the factored equivalent of x2 - 7x + 12.
2.
Since the leading coefficient is 1, look at the signs of the coefficient of x and the constant term.
In this case, the coefficient of x, 7, is positive and the constant term, 10, is positive. So, find the
positive factors of the constant term (think back to getting the middle term when using FOIL to
multiply two binomials).
The constant term can be factored into 1 × 10 or 2 × 5.
Choose the factors of the constant term that when added together give the coefficient of x. This
is usually a trial and error method until the correct combination is found.
In this case, 7 is obtained when 10 is factored as 2 × 5.
Thus, the quadratic will factor into the form (x + p)(x + q) where p and q are the factors of the
constant term.
x2 + 7x + 10 = (x + 5)(x + 2)
3.
The expression is in the form a2x2 + 2abx + b2, where a = 3 and b = 7. Therefore, it can be
factored as a perfect square trinomial.
a2x2 + 2abx + b2 = (ax + b)2
9x2 + 42x + 49 = (3x + 7)2
4.
The expression is in the form a2x2 + 2abx + b2, where a = 2 and b = 3. Therefore, it can be
factored as a perfect square trinomial.
a2x2 + 2abx + b2 = (ax + b)2
4x2 + 12x + 9 = (2x + 3)2
5.
The polynomial shown is a difference of two squares, so it can be factored using the formula
below.
x2 - y2 = (x + y)(x - y)
In this case, y = 3. Use the formula to factor the polynomial.
x2 - 9 = (x + 3)(x - 3)
6.
First, factor out 2, then factor using the difference of two squares.
2x2 - 8 = 2(x2 - 4)
= 2(x + 2)(x - 2)
7.
First, find the greatest common factor of the two coefficients.
Since the greatest common factor of 2 and 5 is 1, a coefficient other than 1 cannot be factored
out of the polynomial.
Next, find the greatest common factor of the variables.
The greatest common factor of x2 and x is x. Factor this out of the polynomial.
2x2 + 5x = x(2x + 5)
8.
First, find the greatest common factor of the two coefficients.
The greatest common factor of -25 and -40 is -5. Factor this out of the polynomial.
-25x2 - 40x = -5(5x2 + 8x)
Next, find the greatest common factor of the variables.
The greatest common factor of x2 and x is x. Factor this out of the polynomial.
-5(5x2 + 8x) = -5x(5x + 8)
9.
Any polynomial in the form x2 + bx + c, where b and c are positive, can be factored using the
formula below.
x2 + (m + n)x + mn = (x + m)(x + n)
In this case, the sum of m and n is equal to b, or 5, and the product of m and n is equal to c, or
4. Find values of m and n that satisfy these conditions.
1+4=5
1·4=4
Substitute m = 1 and n = 4 into the formula to factor the polynomial.
x2 + 5x + 4 = (x + 1)(x + 4)
10.
Start by writing the quadratic in the form ax2 + bx + c.
In this case, a = 2, b = 9, and c = 4.
So, a can be factored into 1 × 2.
Then, c can be factored into 1 × 4 or 2 × 2.
Since all of the coefficients of the quadratic are positive, choose factors of a and c that when one
factor of a is multiplied by one factor of c and that is added to the other factor of a multiplied by
the other factor of c the result is b. This is usually a trial and error method until the correct
combination is found.
In this case, the result is b when a is factored as 1 × 2 and c is factored as 1 × 4 since
1×1+2×4=9
Thus, the quadratic will factor into the form (x + p)(2x + q), where p and q are the factors of c.
Since 1 was multiplied by 1, and 2 was multiplied by 4 (think about the inside and outside terms
of FOIL),
p = 4 and q = 1.
2x2 + 9x + 4 = (x + 4)(2x + 1)
Therefore, 2x + 1 is a factor of the polynomial.
11.
First, factor out the greatest common factor of the coefficients. The greatest common factor of
0.6, -6, and 9.6 is 0.6.
0.6x2 - 6x + 9.6 = 0.6(x2 - 10x + 16)
Next, factor the resulting trinomial.
0.6(x2 - 10x + 16) = 0.6(x - 2)(x - 8)
12.
The polynomial 5x2 + 27x + 10 is in the form ax2 + bx + c, where a = 5, b = 27, and c = 10.
Find the factors of a and c.
5: 1, 5
10: 1, 2, 5, 10
Form two pairs of factors from one factor of a and one factor of c, so that when the factors are
multiplied and then added together (because c is positive), they equal b.
(5 · 5) + (1 · 2) = 27
Use these numbers to form the factors of the polynomial. Because b is positive, both factors will
have addition signs.
(5x + 2)(x + 5)
Therefore, 5x + 2 is a factor of 5x2 + 27x + 10.
13.
The expression is in the form a2x2 + 2abx + b2, where a = 2 and b = 5. Therefore, it can be
factored as a perfect square trinomial.
a2x2 + 2abx + b2 = (ax + b)2
4x2 + 20x + 25 = (2x + 5)2
First, find the greatest common factor of the two coefficients.
14.
The greatest common factor of -30 and -36 is -6. Factor this out of the polynomial.
-30x2 - 36x = -6(5x2 + 6x)
Next, find the greatest common factor of the variables.
The greatest common factor of x2 and x is x. Factor this out of the polynomial.
-6(5x2 + 6x) = -6x(5x + 6)
15.
The expression is in the form a2x2 + 2abx + b2, where a = 5 and b = 4. Therefore, it can be
factored as a perfect square trinomial.
a2x2 + 2abx + b2 = (ax + b)2
25x2 + 40x + 16 = (5x + 4)2