Question

1. teddy factored the polynomial below. his work is shown. did teddy make a mistake? if so, explain how he could correct his work with at least one complete sentence. problem: $32a^{5}b^{4} - 48a^{4}b^{3} + 80a^{3}b^{2}$ answer: $8a^{3}b^{2}(4a^{2}b^{2} - 6ab + 10)$

Explanation:

Step1: Identify GCF of coefficients

Find GCF of 32, 48, 80:
Factors of 32: $1,2,4,8,16,32$
Factors of 48: $1,2,3,4,6,8,12,16,24,48$
Factors of 80: $1,2,4,5,8,10,16,20,40,80$
GCF = $16$

Step2: Identify GCF of $a$-terms

For $a^5, a^4, a^3$, GCF = $a^3$

Step3: Identify GCF of $b$-terms

For $b^4, b^3, b^2$, GCF = $b^2$

Step4: Compute full GCF

Full GCF = $16a^3b^2$

Step5: Verify Teddy's factoring

Teddy used $8a^3b^2$ as the GCF. Divide each term by the correct GCF:
$\frac{32a^5b^4}{16a^3b^2}=2a^2b^2$
$\frac{-48a^4b^3}{16a^3b^2}=-3ab$
$\frac{80a^3b^2}{16a^3b^2}=5$

Answer:

Yes, Teddy made a mistake. He used an incomplete greatest common factor (GCF): he used $8a^3b^2$ instead of the full GCF $16a^3b^2$. To correct his work, he should factor out $16a^3b^2$ instead, resulting in the fully factored form $16a^3b^2(2a^2b^2 - 3ab + 5)$.