Problem 58 - Introductory Algebra for College Students

Question

Factor completely. $$y^{4}-12 y^{3}+35 y^{2}$$

Step-by-Step Solution

Answer

The completely factored form of the polynomial is \(y^{2}(y-5)(y-7)\).

Step 1: Identify and factor out common terms

Notice that each term in the polynomial \(y^{4}-12 y^{3}+35 y^{2}\) includes \(y^2\). Therefore, we can factor out \(y^2\) from each term. The factorised expression will be \(y^{2}(y^{2}-12 y+35)\).

Step 2: Factor the quadratic

In factoring a quadratic expression, look for two numbers that multiply to the constant term (35) and add to the coefficient of the \(y\) term (-12). Those numbers are -5 and -7, as \(-5*-7 = 35\) and \(-5-7 = -12\). Therefore, the quadratic \(y^{2}-12 y+35\) can be factored as \((y-5)(y-7)\). Substitute this back into the original factored expression \(y^{2}(y-5)(y-7)\).

Step 3: Present the final solution

The completely factored form of the polynomial \(y^{4}-12 y^{3}+35 y^{2}\) is \(y^{2}(y-5)(y-7)\).

Key Concepts

Quadratic Expression

A quadratic expression is a type of polynomial, specifically a polynomial of degree 2. This means that the highest power of the variable, typically written as \( y \), is 2. The general form of a quadratic expression is \( ay^2 + by + c \), where \( a \), \( b \), and \( c \) are coefficients. In the polynomial \( y^{4}-12 y^{3}+35 y^{2} \), after factoring out the common factor, we are left with the quadratic expression \( y^2 - 12y + 35 \).

Understanding how to factor quadratic expressions is crucial in simplifying polynomials and finding their roots. To factor a quadratic expression, we look for two numbers that multiply to give the constant term, \( c \), and add up to give the middle coefficient, \( b \). Mastery of factoring will help you solve quadratic equations more easily and understand other complex algebraic concepts.

Common Factor

The common factor of a polynomial is a term or number that can be evenly divided from all terms within the expression. Identifying and removing the common factor is often the first step in the factoring process. For the polynomial \( y^{4}-12 y^{3}+35 y^{2} \), each term includes \( y^2 \), which means \( y^2 \) is the common factor. By factoring out \( y^2 \), we simplify the polynomial to \( y^2(y^2 - 12y + 35) \).

Finding the common factor makes the polynomial easier to handle and often simplifies subsequent calculations. Always check for a common factor at the beginning of your factoring process. It reduces mistakes and clarifies the expression, setting a foundation for further simplification.

Factored Form

The factored form of a polynomial is its expression as a product of simpler polynomials or terms. For the problem \( y^{4}-12 y^{3}+35 y^{2} \), once we factor out the common factor \( y^2 \), we are left with a quadratic that needs factoring. By finding the numbers \( -5 \) and \( -7 \), which multiply to 35 and add to -12, we can express the quadratic as \((y - 5)(y - 7)\). This gives the completely factored form \( y^2(y - 5)(y - 7) \).

Expressing a polynomial in its factored form is important for solving equations, graphing, and simplifying expressions. It displays the zeros or roots of the polynomial, showing the values of the variable where the expression equals zero. Understanding factored form allows for clearer insights into the behavior of polynomial functions.