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Chapter 5: Problem 67

Factor polynomial. \(z^{10}-4 z^{9} y-21 z^{8} y^{2}\)

Short Answer

Expert verified
\( z^8 (z - 7y)(z + 3y) \)

Step by step solution

01

Identify the Greatest Common Factor (GCF)

Look for the greatest common factor in the polynomial terms. The terms in the polynomial are: \( z^{10} \), \( -4z^{9}y \), and \( -21z^{8}y^{2} \). The GCF of these terms is \( z^8 \).
02

Factor Out the GCF

Factor \( z^8 \) out of each term in the polynomial: \[ z^8 (z^2 - 4zy - 21y^2) \]
03

Factor the Quadratic Expression

Now, factor the quadratic expression \( z^2 - 4zy - 21y^2 \). This can be factored by looking for two numbers that multiply to \( -21y^2 \) and add to \( -4zy \): \[ z^2 - 4zy - 21y^2 = (z - 7y)(z + 3y) \]
04

Combine the Factors

Combine the factored form of the polynomial: \[ z^8 (z - 7y)(z + 3y) \]

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Key Concepts

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

Understanding the Greatest Common Factor (GCF)
The Greatest Common Factor (GCF) is very helpful when factoring polynomials. To find the GCF in a polynomial, we look for the largest factor that is common to all terms in the polynomial. For example, in the polynomial given, the terms are \( z^{10} \), \( -4z^{9}y \), and \( -21z^{8}y^{2} \). When we examine these terms carefully, we see that they all contain at least \( z^8 \).
So, the GCF is \( z^8 \).
Factoring out the GCF simplifies the polynomial significantly. This first factoring step reduces the complexity of the entire expression, making it easier to work with.
Factoring Quadratic Expressions
Once we factor out the GCF, we often encounter a simpler quadratic expression that still needs to be factored. Quadratic expressions may look like \( ax^2 + bx + c \). Our example is already in this form, where \( a = 1 \), \( b = -4 \), and \( c = -21y^2 \).
To factor this, we look for two numbers that both:
  • Multiply to give us the constant term (\( -21y^2 \)), and
  • Add up to give the coefficient of the middle term (\( -4z \)).
After examining pairs of factors of \( -21y^2 \), we find that \( -7 \) and \( +3 \) work because \( -7 \times 3 = -21 \) and \( -7 + 3 = -4 \). Hence, the quadratic \( z^2 - 4zy - 21y^2 \) can be factored as \( (z - 7y)(z + 3y) \).
Polynomial Factoring Steps Simplified
Let's break down the steps to factor polynomials, as we saw in the example:
  • Step 1: Find the Greatest Common Factor (GCF) of all terms. For our polynomial, that's \( z^8 \).
  • Step 2: Factor the GCF out of the polynomial. This transforms \( z^{10} - 4z^{9}y - 21z^{8}y^{2} \) to \( z^8(z^2 - 4zy - 21y^2) \).
  • Step 3: Factor the remaining quadratic expression. In our case, \( z^2 - 4zy - 21y^2 \) factors to \( (z - 7y)(z + 3y) \).
  • Step 4: Combine all factors. Our final answer is \( z^8(z - 7y)(z + 3y) \).
These steps help simplify the polynomial, making it much easier to work with or solve for roots.

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