Problem 21 Factor each polynomial. $$ k^{... [FREE SOLUTION]

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Chapter 5: Problem 21

Factor each polynomial. $$ k^{2}-6 k+9 $$

Short Answer

Expert verified

\[ k^{2}-6k+9 = (k-3)^{2} \]

Step by step solution

Identify the Polynomial

Analyze the given polynomial: \[ k^{2}-6k+9 \]

Recognize the Quadratic Form

Observe that the polynomial is a quadratic expression of the form \[ ax^{2} + bx + c \] Here, $a=1$, $b=-6$, and $c=9$.

Find Factors that Multiply to `c` and Sum to `b`

Look for two numbers that multiply to $9$ (the constant term) and add up to $-6$ (the coefficient of the linear term). These two numbers are $-3$ and $-3$ because \[ (-3) \times (-3) = 9 \] and \[ (-3) + (-3) = -6 \].

Rewrite the Quadratic as a Product of Binomials

Express the quadratic polynomial as a product of two binomials: \[ k^{2}-6k+9 = (k-3)(k-3) \]

Consolidate the Expression

Recognize that the expression can be written as the square of a binomial: \[ (k-3)^{2} \]

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Key Concepts

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

Quadratic Expressions

A quadratic expression is a polynomial of degree 2, which means its highest power of the variable is squared. The general form of a quadratic expression is \[ ax^{2} + bx + c \]where:

$a$ is the coefficient of the squared term.
$b$ is the coefficient of the linear term.
$c$ is the constant term.

In our example, \[ k^{2} - 6k + 9 \]we have $a = 1$, $b = -6$, and $c = 9$. Understanding the quadratic expression's structure helps us to identify the necessary steps for factoring it.

Polynomial Factoring

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials. For quadratic polynomials, this typically means expressing them as a product of two binomials. The key steps are:

Identify the quadratic form: Analyze the given polynomial \[ ax^{2} + bx + c \]
Find two numbers that multiply to the constant term $c$ and add up to the linear coefficient $b$.
Rewrite the quadratic expression as the product of two binomials.

For example: \[ k^{2} - 6k + 9 \] is factored by finding two numbers that multiply to 9 (which are -3 and -3) and add up to -6. We can then write the quadratic as $ (k-3)(k-3) $ or $ (k-3)^{2} $.

Binomials

A binomial is a polynomial that has exactly two terms. Binomials can often be factors of quadratic expressions. In the context of our quadratic \[ k^{2} - 6k + 9 \], we identified that it can be expressed as the product of the binomials $ (k-3)(k-3) $. Recognizing binomial patterns is crucial:

It simplifies the process of factoring quadratics.
It helps in understanding the product of sums and differences.
It assists in recognizing perfect square trinomials (like our current example).

By learning to notice these patterns, solving quadratic equations becomes a more straightforward process. In our case, because both binomials were identical, we could further simplify it to $ (k-3)^{2} $.

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91影视

Factor each polynomial. $$ k^{2}-6 k+9 $$

Short Answer

Step by step solution

Identify the Polynomial

Recognize the Quadratic Form

Find Factors that Multiply to `c` and Sum to `b`

Rewrite the Quadratic as a Product of Binomials

Consolidate the Expression

Key Concepts

Quadratic Expressions

Polynomial Factoring

Binomials

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