Problem 80 Factor each trinomial of the for... [FREE SOLUTION]

Chapter 7: Problem 80

Factor each trinomial of the form \(x^{2}+b x+c\). \(y^{2}-5 y+6\)

Short Answer

Expert verified

The factored form is \((y - 2)(y - 3)\).

Step by step solution

- Identify the coefficients

In the trinomial of the form \(y^{2} + by + c\), identify the values of \(b\) and \(c\). Here, \(b = -5\) and \(c = 6\).

- Find two numbers that multiply to \(c\) and add to \(b\)

Find two numbers that multiply to \(c = 6\) and add up to \(b = -5\). These numbers are -2 and -3, because \(-2 \times -3 = 6\) and \(-2 + -3 = -5\).

- Write the factored form

Rewrite the trinomial \(y^{2} - 5y + 6\) as the product of two binomials. Using the numbers found in the previous step, \(y^{2} - 5y + 6 = (y - 2)(y - 3)\).

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Key Concepts

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

algebra

Algebra is the field of mathematics where we use symbols and letters to represent numbers and expressions. It's like a universal language to solve various kinds of problems. Here, we look at how to manipulate and solve equations. When factoring trinomials such as \( y^2 - 5y + 6 \), algebra helps us find simpler expressions that represent the same relationship. Remember that understanding basic operations is crucial. Practicing algebra regularly will improve your skill and confidence.

quadratic equations

A quadratic equation is any equation that can be rearranged in standard form to \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The equation \( y^2 - 5y + 6 \) falls under this category. Quadratic equations often have two solutions and can be solved by factoring, using the quadratic formula, or completing the square. In our case, we used factoring, which turned the quadratic expression into \( (y-2)(y-3) \). This means the values \( y=2 \) and \( y=3 \) are the solutions to the equation.

polynomial factorization

Polynomial factorization involves breaking down a polynomial into the product of simpler polynomials. The goal is to take complex expressions and simplify them. For the trinomial \( y^2 - 5y + 6 \), we want to express it as a product of binomials \( (y - A)(y - B) \). By finding two numbers that multiply to \( c \) (6 in this case) and add up to \( b \) (-5), we were able to rewrite it effectively. Factorization is a key skill because it often reveals solutions and simplifies many algebraic processes.

binomials

Binomials are polynomials with exactly two terms. In the trinomial \( y^2 - 5y + 6 \), we rewrote it as a product of two binomials: \( (y - 2)(y - 3) \). Each binomial represents a simpler part of the original equation. Understanding binomials is crucial because they often serve as building blocks in algebra. Working with binomials helps to simplify complex expressions and solve equations more easily. Practicing with binomials and trinomials will enhance your problem-solving skills in algebra.

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91影视

Factor each trinomial of the form \(x^{2}+b x+c\). \(y^{2}-5 y+6\)

Short Answer

Step by step solution

- Identify the coefficients

- Find two numbers that multiply to \(c\) and add to \(b\)

- Write the factored form

Key Concepts

algebra

quadratic equations

polynomial factorization

binomials

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