Problem 25 Factor each polynomial. $$ 1... [FREE SOLUTION]

Chapter 5: Problem 25

Factor each polynomial. $$ 12 y^{2}+y-1 $$

Short Answer

Expert verified

The factorization is $ (4y - 1)(3y + 1) $.

Step by step solution

Identify the polynomial

The given polynomial is $ 12y^2 + y - 1 $. This is a quadratic polynomial of the form $ a y^2 + b y + c $. Here, $ a = 12 $, $ b = 1 $, and $ c = -1 $.

Find two numbers that multiply to $ a \times c $ and add up to $ b $

Calculate $ a \times c = 12 \times -1 = -12 $. Now, find two numbers that multiply to -12 and add up to 1 (the value of $ b $). These numbers are 4 and -3, because $ 4 \times -3 = -12 $ and $ 4 + (-3) = 1 $.

Rewrite the middle term

Rewrite the polynomial by splitting the middle term using the numbers found in Step 2. The polynomial becomes: \[ 12y^2 + 4y - 3y - 1 \].

Factor by grouping

Group terms to factor by grouping: \[ (12y^2 + 4y) + (-3y - 1) \]. Factor out the greatest common factors from each group: \[ 4y(3y + 1) - 1(3y + 1) \].

Factor out the common binomial factor

Factor out the common factor $ 3y + 1 $: \[ (4y - 1)(3y + 1) \]. Therefore, the factorization of $ 12y^2 + y - 1 $ is $ (4y - 1)(3y + 1) $.

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Key Concepts

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

headline of the respective core concept

Quadratic polynomials are expressions of the form $ay^2 + by + c$, where $a$, $b$, and $c$ are constants. In the given exercise, the quadratic polynomial is $12y^2 + y - 1$. Understanding the format helps in solving the problem. Every quadratic polynomial creates a parabola when graphed, which is a symmetric curve.

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Factoring by grouping is a method used to factor polynomials when handling four terms. Here's how it works:
1. Split the middle term into two parts. For example, the term $y$ in $12y^2 + y - 1$ splits into $4y$ and $-3y$.
2. Rearrange and group terms: $12y^2 + 4y - 3y - 1$.
3. Factor out the greatest common factor from each group: $4y(3y + 1) - 1(3y + 1)$.
By grouping, you simplify the polynomial into smaller, more manageable pieces.

headline of the respective core concept

Binomial factors are expressions that have two terms. After factoring by grouping the polynomial $12y^2 + y - 1$, you notice both groups contain a common binomial factor: $3y + 1$.
To finish, you factor out the common binomial factor, leading to the simplified product $ (4y - 1)(3y + 1) $.
Thus, the original quadratic polynomial is factored into two binomial factors.

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91影视

Factor each polynomial. $$ 12 y^{2}+y-1 $$

Short Answer

Step by step solution

Identify the polynomial

Find two numbers that multiply to \( a \times c \) and add up to \( b \)

Rewrite the middle term

Factor by grouping

Factor out the common binomial factor

Key Concepts

headline of the respective core concept

headline of the respective core concept

headline of the respective core concept

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