Problem 31 Factor completely. If the polyno... [FREE SOLUTION]

Chapter 5: Problem 31

Factor completely. If the polynomial cannot be factored, write prime. \(y^{2}-8 y+15\)

Short Answer

Expert verified

The polynomial \( y^2 - 8y + 15 \) factors to \( (y - 3)(y - 5) \).

Step by step solution

Identify the Form

The given polynomial is a quadratic of the form \[ ax^2 + bx + c \]. For this exercise, identify the coefficients: \[ a = 1, b = -8, c = 15 \].

Find Factors of Constant Term

List pairs of factors of the constant term \( c \) that is 15. Pairs include: (1, 15), (3, 5), (-1, -15), and (-3, -5).

Identify Pair That Sums to Middle Term

Among the pairs, find two numbers whose product equals 15 and sum equals the middle coefficient, -8. The pair (-3, -5) satisfies this since \[ -3 + (-5) = -8 \] and \[ -3 \times -5 = 15 \].

Write Factored Form

Express the polynomial as a product of binomials using the identified pair: \[ y^2 - 8y + 15 = (y - 3)(y - 5) \].

Verify Factored Form

Expand \[ (y - 3)(y - 5) \] to verify: \[ y^2 - 5y - 3y + 15 = y^2 - 8y + 15 \]. The factored form is correct.

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Key Concepts

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

quadratic equations

A quadratic equation is a polynomial equation of degree 2. It takes the general form: \( ax^2 + bx + c = 0 \).
Here,

\( a \) is the coefficient of \( x^2 \),
\( b \) is the coefficient of \( x \),
\( c \) is the constant term.

In the given exercise, we have the quadratic equation:
\( y^2 - 8y + 15 \).
This means:

\( a = 1 \),
\( b = -8 \),
and \( c = 15 \).

Our goal is to factor this equation completely, if possible.
This involves breaking down the quadratic into a product of two binomials.

polynomial factorization

Polynomial factorization is the process of expressing a polynomial as a product of its factors.
For quadratic equations, this often involves writing them as products of binomials.
Let's break down our example. First, we list the factors of the constant term \( c = 15 \).

These pairs are: (1, 15), (3, 5), (-1, -15), and (-3, -5).

Next, we need to find the pair of factors that add up to the middle coefficient \( b = -8 \).
Among these pairs, we see that (-3) and (-5) work because:

\( -3 + (-5) = -8 \)
and \( -3 \times -5 = 15 \).

We then express the polynomial using these numbers:
\[ y^2 - 8y + 15 = (y - 3)(y - 5) \].

factored form verification

Once we have factored the quadratic equation, it鈥檚 important to verify that the factorization is correct.
We do this by expanding the factored form and checking if we get back the original polynomial.
Starting with our factors:
\( (y - 3)(y - 5) \)
We expand it by applying the distributive property (FOIL Method):

\( y \times y = y^2 \)
\( y \times -5 = -5y \)
\( -3 \times y = -3y \)
\( -3 \times -5 = 15 \)

Combining these terms gives:
\[ y^2 - 5y - 3y + 15 = y^2 - 8y + 15 \].
The expanded form matches the original polynomial, confirming that
\( y^2 - 8y + 15 = (y - 3)(y - 5) \) is the correct factorization.

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91影视

Factor completely. If the polynomial cannot be factored, write prime. \(y^{2}-8 y+15\)

Short Answer

Step by step solution

Identify the Form

Find Factors of Constant Term

Identify Pair That Sums to Middle Term

Write Factored Form

Verify Factored Form

Key Concepts

quadratic equations

polynomial factorization

factored form verification

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