Question

polynomials and factoring — review assignm

1. factor completely.

(a) $3h^2 + 12h - 36$

(b) $8k^2 - 24k + 18$

(c) $x^2 - 36$

(d) $25 - 64b^2$

Explanation:

Part (a)

Step1: Factor out GCF

The greatest common factor (GCF) of \(3h^2\), \(12h\), and \(-36\) is \(3\). Factor out \(3\):
\(3h^2 + 12h - 36 = 3(h^2 + 4h - 12)\)

Step2: Factor the quadratic

Factor \(h^2 + 4h - 12\). We need two numbers that multiply to \(-12\) and add to \(4\). These numbers are \(6\) and \(-2\):
\(h^2 + 4h - 12 = (h + 6)(h - 2)\)

Step3: Combine

Putting it together, the completely factored form is:
\(3(h + 6)(h - 2)\)

Part (b)

Step1: Factor out GCF

The GCF of \(8k^2\), \(-24k\), and \(18\) is \(2\). Factor out \(2\):
\(8k^2 - 24k + 18 = 2(4k^2 - 12k + 9)\)

Step2: Factor the quadratic

The quadratic \(4k^2 - 12k + 9\) is a perfect square trinomial: \((2k - 3)^2\) (since \((2k)^2 = 4k^2\), \(2(2k)(-3) = -12k\), and \((-3)^2 = 9\)).

Step3: Combine

Thus, the completely factored form is:
\(2(2k - 3)^2\)

Part (c)

Step1: Recognize difference of squares

The expression \(x^2 - 36\) is a difference of squares: \(a^2 - b^2 = (a + b)(a - b)\), where \(a = x\) and \(b = 6\) (since \(6^2 = 36\)).

Step2: Apply formula

Using the difference of squares formula:
\(x^2 - 36 = (x + 6)(x - 6)\)

Part (d)

Answer:

s:
(a) \(\boldsymbol{3(h + 6)(h - 2)}\)
(b) \(\boldsymbol{2(2k - 3)^2}\)
(c) \(\boldsymbol{(x + 6)(x - 6)}\)
(d) \(\boldsymbol{(5 + 8b)(5 - 8b)}\)