Question

question 2 of 10
which of the binomials below is a factor of this trinomial?
$4x^2 + 20x - 24$

a. $x - 4$
b. $x + 1$
c. $x + 4$
d. $x - 1$

Explanation:

Step1: Factor out the GCF

First, factor out the greatest common factor (GCF) of the trinomial \(4x^{2}+20x - 24\). The GCF of 4, 20, and -24 is 4. So we have:
\(4(x^{2}+5x - 6)\)

Step2: Factor the quadratic

Now factor the quadratic \(x^{2}+5x - 6\). We need two numbers that multiply to -6 and add to 5. Those numbers are 6 and -1. So we can factor it as:
\(x^{2}+5x - 6=(x + 6)(x - 1)\)

Step3: Combine the factors

Putting it all together, the factored form of the original trinomial is:
\(4x^{2}+20x - 24 = 4(x + 6)(x - 1)\)

Looking at the options, \(x - 1\) (option D) is a factor. Wait, but let's check by plugging in the roots. The roots of the trinomial can be found by setting it equal to zero. From \(4(x + 6)(x - 1)=0\), the roots are \(x=-6\) and \(x = 1\). For a binomial \((x - a)\) to be a factor, \(a\) should be a root. Let's check each option:

• Option A: \(x - 4\), root at \(x = 4\). Plug \(x = 4\) into \(4x^{2}+20x - 24\): \(4(16)+80 - 24=64 + 80 - 24 = 120

eq0\), so not a factor.

• Option B: \(x + 1=(x - (-1))\), root at \(x=-1\). Plug \(x=-1\): \(4(1)+(-20)-24=4 - 20 - 24=-40

eq0\), not a factor.

• Option C: \(x + 4=(x - (-4))\), root at \(x=-4\). Plug \(x=-4\): \(4(16)+(-80)-24=64 - 80 - 24=-40

eq0\), not a factor.

• Option D: \(x - 1\), root at \(x = 1\). Plug \(x = 1\): \(4(1)+20 - 24=4 + 20 - 24=0\), so it is a factor.

Wait, but earlier when we factored, we had \(4(x + 6)(x - 1)\), so \(x - 1\) is a factor. But let's re - check the factoring of the quadratic. Wait, \(x^{2}+5x - 6\): product of -6 and sum of 5. 6 and -1: \(6\times(-1)=-6\) and \(6+(-1)=5\), that's correct. So the factoring is correct.

Wait, but let's check the initial factoring again. The original trinomial is \(4x^{2}+20x - 24\). Dividing by 4: \(x^{2}+5x - 6\). Factoring: \(x^{2}+5x - 6=(x + 6)(x - 1)\), correct. So the factors are \(4\), \(x + 6\), and \(x - 1\). So among the options, \(x - 1\) (option D) is a factor.

Answer:

D. \(x - 1\)