Problem 11 Factor. Check your answer by mul... [FREE SOLUTION]

Chapter 29: Problem 11

Factor. Check your answer by multiplying. $$ 28 x^{6}-12 x^{4}+20 x^{2} $$

Short Answer

Expert verified

The factored form is $4x^2 (7x^4 - 3x^2 + 5)$.

Step by step solution

- Find the Greatest Common Factor (GCF)

Identify the GCF of all terms in the polynomial. The terms are $28x^6$, $-12x^4$, and $20x^2$. The GCF of the coefficients 28, -12, and 20 is 4. The lowest power of $x$ present in each term is $x^2$. Thus, the GCF is $4x^2$.

- Factor out the GCF

Divide each term by the GCF and factor it out: \[ 28x^6 - 12x^4 + 20x^2 = 4x^2 (7x^4 - 3x^2 + 5) \]

- Verify the Factoring by Multiplying

To ensure the factoring is correct, distribute $4x^2$ back through the polynomial: \[ 4x^2 (7x^4 - 3x^2 + 5) = (4x^2)(7x^4) - (4x^2)(3x^2) + (4x^2)(5) = 28x^6 - 12x^4 + 20x^2 \] Since the multiplication reproduces the original polynomial, the factorization is verified.

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Key Concepts

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

Greatest Common Factor

When factoring polynomials, it helps to start by finding the Greatest Common Factor (GCF). The GCF is the largest factor shared by all terms in the polynomial.
Identify the GCF by focusing on the coefficients and variables separately.
For the polynomial given, the terms are 28x鈦�, -12x鈦�, and 20x虏. First, we find the GCF of the coefficients (28, -12, and 20). The common factor is 4.
Next, look at the variables: the lowest power of x in all terms is x虏.
Combining these two, the GCF is 4x虏.
Here is how to break it down step-by-step:

Identify the largest number that divides all coefficients: 4.
Determine the lowest power of the variable common to all terms: x虏.

Hence, for the polynomial 28x鈦� - 12x鈦� + 20x虏, the GCF is 4x虏.

Polynomial Factorization

To factorize a polynomial, you'll divide each term by the GCF and then rewrite it in a simplified form.
After identifying the GCF for 28x鈦� - 12x鈦� + 20x虏 as 4x虏, we factor it out.
This means we divide each term in the polynomial by 4x虏.
Here's the breakdown:

28x鈦� 梅 4x虏 = 7x鈦�
-12x鈦� 梅 4x虏 = -3x虏
20x虏梅 4x虏 = 5

After dividing each term, we rewrite the polynomial as:
28x鈦� - 12x鈦� + 20x虏 = 4x虏 (7x鈦� - 3x虏 + 5).
This simplified form is the factorized version of the original polynomial.
By factoring, we make our polynomial easier to work with for solving equations or finding roots.

Multiplication Verification

To ensure the accuracy of the factorization, we use multiplication verification. This method checks if the original polynomial can be obtained by multiplying back the factors.
Given the factorized polynomial 4x虏 (7x鈦� - 3x虏 + 5), we multiply the terms to verify:
Multiply 4x虏 by each term inside the parentheses:

4x虏 * 7x鈦� = 28x鈦�
4x虏 * (-3x虏) = -12x鈦�
4x虏 * 5 = 20x虏

By distributing 4x虏 through the polynomial, we get:
(4x虏)(7x鈦�) - (4x虏)(3x虏) + (4x虏)(5) = 28x鈦� - 12x鈦� + 20x虏.
The result matches the original polynomial, confirming the factorization is correct.
Multiplication verification is a reliable way to ensure your factorized form is accurate.

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91影视

Factor. Check your answer by multiplying. $$ 28 x^{6}-12 x^{4}+20 x^{2} $$

Short Answer

Step by step solution

- Find the Greatest Common Factor (GCF)

- Factor out the GCF

- Verify the Factoring by Multiplying

Key Concepts

Greatest Common Factor

Polynomial Factorization

Multiplication Verification

One App. One Place for Learning.

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