Problem 116 Factor each expression. \(y^{2... [FREE SOLUTION]

Chapter 7: Problem 116

Factor each expression. \(y^{2}+41 y+40\)

Short Answer

Expert verified

The factored form is \( (y + 1)(y + 40) \).

Step by step solution

- Identify A, B, and C

In the quadratic expression in the form of \( ax^2 + bx + c \), identify the coefficients. Here, \( a = 1 \), \( b = 41 \), and \( c = 40 \).

- Find two numbers that multiply to C and add to B

Look for two numbers that multiply to \( c = 40 \) and add to \( b = 41 \). The numbers 1 and 40 satisfy these conditions because \( 1 \times 40 = 40 \) and \( 1 + 40 = 41 \).

- Write the quadratic as a product of binomials

Express the quadratic expression as a product of two binomials. Using the numbers found in the previous step, the factorized form is: \( (y + 1)(y + 40) \).

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Key Concepts

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

Quadratic Expressions

Quadratic expressions are a type of polynomial with the highest exponent of 2. This means that the expression will include a term like \(ax^2\). A standard quadratic expression looks like \(ax^2 + bx + c\). Here, \(a\), \(b\), and \(c\) are known as the coefficients. They can be any real number. Understanding the structure helps you when factoring the expression. In our example, \(y^2 + 41y + 40\), the quadratic term is \(y^2\), indicating it's a quadratic expression. The trick to factoring is to find pairs of numbers that both add and multiply correctly.

Coefficients

Coefficients are the numerical parts attached to the variables in a polynomial. For example, in the quadratic expression \(y^2 + 41y + 40\), we have coefficients as follows: \(a = 1\), \(b = 41\), and \(c = 40\). Knowing how to identify these values is crucial.
\[ \begin{align*} a \ & = \text{1 (coefficient of } y^2) \ b \ & = 41 \ c \ & = 40 \end{align*} \] These coefficients help you find the two key numbers for factoring. For \(y^2 + 41y + 40\), you want two numbers that multiply to \(c\) and add up to \(b\).

Binomials

A binomial is a polynomial with exactly two terms. When you factor a quadratic expression, you often write it as the product of two binomials. For instance, \( (y + 1)(y + 40) \) are our binomials. Factoring basically rewrites the quadratic expression in a more understandable form.
Start by finding two numbers that multiply to the constant term and add up to the linear coefficient. In our exercise, the numbers are 1 and 40 because they meet both conditions: \(1 \times 40 = 40\) and \(1 + 40 = 41\). Consequently, our quadratic expression \(y^2 + 41y + 40\) factors to \((y + 1)(y + 40)\).

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91影视

Factor each expression. \(y^{2}+41 y+40\)

Short Answer

Step by step solution

- Identify A, B, and C

- Find two numbers that multiply to C and add to B

- Write the quadratic as a product of binomials

Key Concepts

Quadratic Expressions

Coefficients

Binomials

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