Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ 15+2 x-x^{2} $$

Short answer

\(-(x - 5)(x + 3)\)

Step-by-step solution

Write the polynomial in standard form

Rewrite the given polynomial in standard form from highest to lowest degree of x. The polynomial given is: \(15 + 2x - x^2\) In standard form, this becomes: \(-x^2 + 2x + 15\)

Factor out the negative sign (if any)

To make factoring easier, you can factor out the negative sign from each term: \(-x^2 + 2x + 15\) This becomes: \(-(x^2 - 2x - 15)\)

Identify the trinomial for factoring

Now, focus on factoring the quadratic trinomial inside the parentheses: x^2 - 2x - 15

Find two numbers that multiply to -15 and add to -2

To factor the trinomial \(x^2 - 2x - 15\), find two numbers that multiply to the constant term (-15) and add up to the linear coefficient (-2). These numbers are 3 and -5.

Decompose the middle term

Use the numbers found in the previous step to decompose the middle term: x^2 - 2x - 15 = x^2 - 5x + 3x - 15

Factor by grouping

Group the terms in pairs and factor each pair separately: (x^2 - 5x) + (3x - 15) = x(x - 5) + 3(x - 5)

Factor out the common factor

Factor out the common binomial factor from the groups: = (x - 5)(x + 3)

Reinsert the negative sign

Don't forget the negative sign we factored out earlier. Reinsert it to get the final factored form: The given polynomial \(15 + 2x - x^2\) factors to: \(-(x - 5)(x + 3)\)

Key concepts

Standard Form

Understanding the standard form of a polynomial is essential for factoring. A polynomial is in standard form when its terms are written in descending order of their degrees. For example, consider a polynomial like this:\[ 15 + 2x - x^2 \]. First, identify the highest-degree term, which is \( -x^2 \), and rewrite the polynomial so it reads \( -x^2 + 2x + 15 \).
This arrangement helps you more easily see the structure of the polynomial for further steps.

Factoring Trinomials

Factoring trinomials is a common task in algebra. A trinomial is a polynomial with three terms, and factoring it involves rewriting it as the product of two binomials.
For the trinomial \( x^2 - 2x - 15 \), you need to find two numbers that multiply to the constant term (-15) and add up to the coefficient of x (-2).
Here, those numbers are 3 and -5 because \( 3 \times -5 = -15 \) and \( 3 + (-5) = -2 \). These numbers help you decompose the middle term and simplify the trinomial to two binomials: \( (x - 5)(x + 3) \).

Factoring by Grouping

Factoring by grouping is an efficient method for polynomials with four terms. This process involves grouping terms in pairs and factoring out common factors from each pair.
Consider the polynomial \( x^2 - 5x + 3x - 15 \). Group the terms as pairs: \((x^2 - 5x)\) and \((3x - 15)\).
Factor out the common factors from each pair: \( x(x - 5) \) and \( 3(x - 5) \).
You'll notice that \( x - 5 \) is a common factor, allowing you to rewrite the polynomial as \( (x - 5)(x + 3) \).

Quadratic Equations

Quadratic equations are polynomials of degree two, typically written in the form \( ax^2 + bx + c = 0 \). They are important in algebra and can be solved by factoring, completing the square, or using the quadratic formula.
When you have a quadratic expression like \( -x^2 + 2x + 15 \), putting it into standard form (\( x^2 - 2x - 15 \)) makes it easier to factor. Factoring breaks it down into simpler components (\( (x - 5)(x + 3) \)), which can then be tackled as more manageable equations.
This method shows the importance of quadratic equations in solving polynomial problems.