Problem 44 Factor completely. If the polyno... [FREE SOLUTION]

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Beginning Algebra

Margaret L. Lial, John Hornsby

$Math Studyset 91影视 Explanations$ Math

11 Edition

Chapter 6: Problem 44

Factor completely. If the polynomial cannot be factored, write prime. $$ -39+10 x+x^{2} $$

Short Answer

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Step by step solution

Write the Polynomial in Standard Form

Arrange the polynomial in descending order of the degree of the terms: This gives us: Replace incorrect symbols with correct ones as it's critical to maintain latex Correct form This gives us: Update formula rewrite polynomial in standard form Fix symbols update example Fix issues with bullet points Remove text This gives us: The correct formula Fix format and extra spaces

Solve Using the Quadratic Formula

Quadratic formula is given as Update the correct formula The this can be done by solving the quadratic formula Adjust for correct latex formula Replace text not relevant with correct latex version This gives us partial quadratics

Check Factorization

Re-arrange to create a type of correct quadratic factor formula. To rewrite formula Replace correct quadratic factor symbols This gives us Fix issues

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Key Concepts

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

Polynomial Factorization

Polynomial factorization is an important skill in algebra. It involves breaking down a polynomial into simpler 'factors' that, when multiplied together, yield the original polynomial.

For example, to factor the polynomial given in the exercise, $x^2 + 10x - 39$, you need to find two binomials that multiply to give this polynomial. This process often involves the following steps:

Factoring out the greatest common factor (GCF), if there is one.
Rewriting the polynomial in a standard form (descended order of powers).
Using techniques like grouping, quadratic formula or special factoring formulas.

In our case, the polynomial didn't have a GCF, so we used the quadratic formula to solve it, which eventually helped us find its factors.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. A quadratic equation is of the form $ax^2 + bx + c = 0$. The formula for solving this is:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

To factor our polynomial $x^2 + 10x - 39$, we used the quadratic formula where:

$a = 1$
$b = 10$
$c = -39$

Plugging these values into the quadratic formula gives us the roots of the equation. These roots are then used to form the factors.
After solving, we found that $x = 3$ and $x = -13$, meaning our polynomial factors into $ (x - 3)(x + 13) $.

Standard Form of a Polynomial

Putting a polynomial in standard form is often the first step in most algebraic manipulations. The standard form of a polynomial means arranging its terms in descending order of their degrees. For our exercise, the given polynomial is $-39 + 10x + x^2$.

Rewriting this in standard form, we get $ x^2 + 10x - 39 $.

This rearrangement helps simplify the process of factoring or applying the quadratic formula as it clearly shows the highest power first and makes identifying the coefficients easier.
Always remember:

Write the terms in descending order of their exponents.
Check for missing terms or coefficients to avoid mistakes.

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91影视

Factor completely. If the polynomial cannot be factored, write prime. $$ -39+10 x+x^{2} $$

Short Answer

Step by step solution

Write the Polynomial in Standard Form

Solve Using the Quadratic Formula

Check Factorization

Key Concepts

Polynomial Factorization

Quadratic Formula

Standard Form of a Polynomial

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