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Chapter 10: Problem 46

Factor out the greatest common factor. $$7 y^{4}+9 y^{3}-11 y^{2}$$

Short Answer

Expert verified
y^2(7y^2 + 9y - 11)

Step by step solution

01

Identify the Greatest Common Factor (GCF)

Examine each term in the expression to find the greatest common factor. The terms are: 1. 7y^4 2. 9y^3 3. 11y^2 The common factor for all the terms is y^2.
02

Factor out the GCF from each term

Once the GCF, which is y^2, is identified, factor it out from each term: 7y^4 + 9y^3 - 11y^2 = y^2(7y^2 + 9y - 11)

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Key Concepts

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

greatest common factor
Understanding the concept of the greatest common factor (GCF) is crucial in algebra. The GCF is the highest number or term that evenly divides all the given terms. In the example 7y^4, 9y^3, and 11y^2, we look for the highest power of y that is present in all terms. Here, each term contains at least y^2. Thus, y^2 = GCF. Identifying this helps us simplify expressions by factoring out the common term.
algebra
Algebra involves working with symbols and numbers to solve problems. Here, we work with polynomial expressions. By factoring out the GCF, we transform the expression into a simpler form. In the expression 7y^4 + 9y^3 - 11y^2, algebraic manipulation helps break it down. We factor out y^2 to get y^2(7y^2 + 9y - 11). This simplification aids in further solving or understanding the polynomial.
polynomial equations
Polynomial equations involve terms with variables raised to whole number exponents. These equations can often be simplified by factoring. For example, identifying the GCF in a polynomial like 7y^4 + 9y^3 - 11y^2 makes the equation easier to manage. After factoring out y^2, we are left with a simpler polynomial: y^2(7y^2 + 9y - 11). This step-by-step simplification using factoring is foundational in solving polynomial equations efficiently.

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Most popular questions from this chapter

Simplify. Write the answer with positive exponents only. $$2 x^{-2} y^{-3}$$

Simplify the expression. $$\left(\frac{9 u}{t}\right)^{2}$$

Subtract the polynomials. $$\left(12 c^{2}+6 c-3\right)-\left(4 c^{2}-3 c+8\right)$$

Write the opposite of the polynomial. $$9 x^{2}-2 x-3$$

$$\text { Fill in the blanks. }\left(y^{2}\right)^{7}=y \square \cdot \square=y^{14}$$
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