Problem 24 Check by multiplying. $$ 9 b... [FREE SOLUTION]

Chapter 10: Problem 24

Check by multiplying. $$ 9 b^{3}-54 b^{2}+9 b $$

Short Answer

Expert verified

The polynomial is factored as $9b(b^2 - 6b + 1)$.

Step by step solution

Factor out the greatest common factor (GCF)

We first identify the greatest common factor among the terms in the polynomial. The terms are $9b^3$, $-54b^2$, and $9b$. The common factor here is $9b$. Thus, we factor $9b$ out of each term: \[9b(b^2 - 6b + 1)\]

Verify by multiplying

To ensure correctness, we multiply back the factored expression to check if it matches the original polynomial. Distribute $9b$ across each term inside the parentheses: \[9b(b^2 - 6b + 1) = 9b \cdot b^2 - 9b \cdot 6b + 9b \cdot 1\]Simplify each term: \[= 9b^3 - 54b^2 + 9b\]This matches the original polynomial, confirming that our factoring is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Factor

The greatest common factor (GCF) is an essential concept when working with polynomials. It helps to simplify polynomial expressions by factoring out the largest integer and variable combination that divides each term evenly.
To find the GCF of a polynomial, break down each coefficient into prime factors and evaluate the smallest power of each variable appearing in all terms. For example, in the polynomial given, the terms are: $9b^3$, $-54b^2$, and $9b$.

First, compare the numerical coefficients: 9, -54, and 9. The GCF is 9. This is because it's the highest number that divides all these coefficients.
Next, look at the variable part: $b^3$, $b^2$, and $b$. The lowest power of $b$ here is $b^1$, thus the GCF includes $b$.

Together, the GCF is $9b$. Factoring it out from each term gives you $9b(b^2 - 6b + 1)$, a simpler expression of the polynomial.

Multiply Polynomials

Multiplying polynomials is a critical skill for verifying factored expressions. After factoring, you should multiply back to check your work. This process involves distributing each term of one polynomial to every term of another.
In our polynomial, you need to multiply $9b$ across the terms inside the parentheses $(b^2 - 6b + 1)$ after factoring:

Start by multiplying $9b$ with $b^2$: $9b \cdot b^2 = 9b^3$.
Next, multiply $9b$ with $-6b$: $9b \cdot (-6b) = -54b^2$.
Finally, multiply $9b$ with $1$: $9b \cdot 1 = 9b$.

Add the products together: $9b^3 - 54b^2 + 9b$. This operation confirms each part has been correctly multiplied, proving that the initial factorization was accurate.

Polynomial Verification

Polynomial verification ensures the factors and the original polynomial match after reverting the factorization. Verification provides a reliable check that no mistakes were made during the factoring process.
Once you have multiplied back your factors, as shown in prior sections, compare the result with the original polynomial.

If they match, the factoring is correct.
If not, re-evaluate each step for errors.

Now consider our example: starting with the factored form $9b(b^2 - 6b + 1)$ and multiplying gives us back the original $9b^3 - 54b^2 + 9b$. This confirms our operation was accurate.
Verification creates confidence, ensuring you've correctly simplified the expression without changing its value.

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91影视

Check by multiplying. $$ 9 b^{3}-54 b^{2}+9 b $$

Short Answer

Step by step solution

Factor out the greatest common factor (GCF)

Verify by multiplying

Key Concepts

Greatest Common Factor

Multiply Polynomials

Polynomial Verification

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