QuestionIdentify the GCF and then factor the GCF out of the expression. $22 a^{2} b^{6} c^{8}+10 a^{9} b^{3}-16 a^{5} b c^{7}$
The GCF is: $\square$
The factored expression is: $\square$

Studdy Solution

STEP 1

1. We are given a polynomial expression with three terms.
2. We need to identify the greatest common factor (GCF) of the terms.
3. We will factor out the GCF from the expression.

STEP 2

1. Identify the GCF of the coefficients and the variables in the expression.
2. Factor the GCF out of the expression.

STEP 3

Identify the GCF of the coefficients $22$ , $10$ , and $-16$ .
- The prime factors of $22$ are $2 \times 11$ . - The prime factors of $10$ are $2 \times 5$ . - The prime factors of $-16$ are $2^4$ .
The GCF of the coefficients is $2$ .

STEP 4

Identify the GCF of the variables in each term:
- For $a$ , the powers are $2$ , $9$ , and $5$ . The GCF is $a^2$ . - For $b$ , the powers are $6$ , $3$ , and $1$ . The GCF is $b^1$ . - For $c$ , the powers are $8$ , $0$ , and $7$ . The GCF is $c^0$ (since $c$ is not present in the second term).
The GCF of the entire expression is $2a^2b$ .

STEP 5

Factor the GCF $2a^2b$ out of each term in the expression:
\[ 22a^2b^6c^8 + 10a^9b^3 - 16a^5bc^7 $
becomes:
\[ 2a^2b(11b^5c^8 + 5a^7b - 8a^3c^7) $
The GCF is: $2a^2b$
The factored expression is: $2a^2b(11b^5c^8 + 5a^7b - 8a^3c^7)$

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