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Chapter 6: Problem 7

Factor completely. If a polynomial is prime, state this. $$ y^{2}+49-14 y $$

Short Answer

Expert verified
The polynomial \( y^2 - 14y + 49 \) can be factored as \( (y - 7)^2 \).

Step by step solution

01

Arrange the polynomial

First, write the polynomial in standard form, which is usually organized by decreasing powers of the variable. The given polynomial is: \[ y^2 - 14y + 49 \]
02

Identify a, b, and c

Identify the coefficients in the quadratic polynomial of the form \[ ay^2 + by + c \]. Here, a = 1, b = -14, and c = 49.
03

Apply the factoring technique

To factor \[ y^2 - 14y + 49 \]directly as a perfect square trinomial, notice that \( y^2 \) is the square of \( y \), and \( 49 \) is the square of \( 7 \). The middle term, \( -14y \), is \(-2 (7)(y)\).
04

Write in the perfect square trinomial format

A perfect square trinomial can be factored as \( (a - b)^2 \). Here, \( y^2 - 14y + 49 \) can be written as \( (y - 7)^2 \).

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Key Concepts

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

quadratic polynomials
Quadratic polynomials are expressions of the form ax^2 + bx + c where a, b, and c are coefficients, and x is the variable. Quadratics are called so because the highest power of the variable is 2 (the term 'quadratic' comes from the Latin word for 'square'). In most cases, this power is denoted by x^2. Quadratic polynomials have several properties useful for their manipulation:
  • The graph of a quadratic polynomial is a parabola.
  • They have at most two roots (solutions) which can be found using the quadratic formula, factoring, or completing the square.
  • They can be represented in standard form, as given (ax^2 + bx + c), vertex form, or factored form.

In our exercise, the polynomial provided is
y^2 - 14y + 49
which is already in standard form.

The first step in solving the exercise involved ensuring the quadratic polynomial is appropriately organized.
perfect square trinomials
Perfect square trinomials are special quadratic polynomials formed by squaring a binomial. They take the form a^2 + 2ab + b^2 or a^2 - 2ab + b^2 Here, the polynomial y^2 - 14y + 49 needs to be checked for this pattern.

The trick is to recognize that y^2 is the square of y and 49 is the square of 7 . The middle term -14y confirms this pattern, as it is -2(7)(y) .

This identification helps in factoring the polynomial in the form (a - b)^2 or (a + b)^2 .
standard form of polynomials
The standard form of polynomials is a way to organize the polynomial by the descending powers of its variable. For any polynomial, writing it in standard form involves arranging the terms from highest to lowest power. For quadratic polynomials, standard form looks like ax^2 + bx + c

Using this approach, the polynomial y^2 - 14y + 49 is already correctly written, enabling us to easily identify the coefficients:
  • a = 1
  • b = -14
  • c = 49

Having the polynomial in standard form allows us to easily recognize patterns such as perfect square trinomials.

After recognizing the pattern in our problem, we can then express it as
(y - 7)^2.

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Most popular questions from this chapter

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ \frac{1}{49}-x^{2} $$

Solve. Round any irrational solutions to the nearest thousandth. $$ x^{2}=\frac{1}{100} $$

Solve. Round any irrational solutions to the nearest thousandth. $$ 3 x^{2}+x-1=0 $$

Factor completely. Assume that variables in exponents represent positive integers. $$ a^{2}+2 a b+b^{2}-c^{2}+6 c-9 $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ 9 a^{4}-25 a^{2} b^{4} $$
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