Problem 53 - Elementary Algebra

Question

For the following problems, factor the polynomials, if possible. $$ a^{2}-9 a+20 $$

Step-by-Step Solution

Answer

Answer: The factored form of the quadratic polynomial \(a^{2}-9 a+20\) is \((a - 4)(a - 5)\).

Step 1: Identify the coefficients and terms

The quadratic polynomial has the form: $$a^{2}-9 a+20.$$ The coefficients are: $$a^2$$ coefficient: 1, $$a$$ coefficient: -9, constant term: 20.

Step 2: Find two numbers that multiply to 20 and add to -9

We need to find two numbers whose product is 20 (the constant term) and whose sum is -9 (the coefficient of the linear term). After trying multiple combinations, we find that the numbers -4 and -5 work, because: $$(-4) \times (-5) = 20$$ and $$(-4) + (-5) = -9$$.

Step 3: Rewrite the polynomial with the new terms

Replace the middle term, -9a, with the new terms, -4a and -5a, so the polynomial becomes: $$a^2 - 4a - 5a + 20$$.

Step 4: Factor the polynomial using the distributive property

Now we can factor the polynomial by grouping the first two terms and the last two terms together. Factor out the greatest common factor from each group: $$a(a - 4) - 5(a - 4)$$.

Step 5: Factor the common term

We can now factor out the common term, (a - 4), from both groupings: $$(a - 4)(a - 5)$$. So, the factored form of the polynomial $$a^{2}-9 a+20$$ is $$(a - 4)(a - 5)$$.

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental concept in algebra that involves breaking down a polynomial into simpler components or "factors." In our given problem, we aim to rewrite the quadratic polynomial \(a^2 - 9a + 20\) as a product of two binomials. This process makes complex expressions easier to solve and understand.

• First, identify the structure of the polynomial, typically written in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients.
• Determine two numbers that multiply to the constant term \(c\) (20 in this case) and add up to the linear coefficient \(b\) (-9 here).
• These numbers, once found, help in rewriting and factoring the polynomial into simpler expressions.

By selecting appropriate numbers, factoring allows us to simplify equations and solve them efficiently.

Quadratic Polynomial

A quadratic polynomial is a type of polynomial where the highest degree of any term is two. The standard form of a quadratic polynomial is expressed as \(ax^2 + bx + c\). In our exercise, the quadratic polynomial given is \(a^2 - 9a + 20\), where

• \(a^2\) is the quadratic term with a leading coefficient of 1,
• -9a is the linear term, and
• 20 is the constant term.

The process of solving equations with quadratic polynomials often involves factoring, which can simplify the equation. This method is particularly useful because it reveals the roots or solutions of the polynomial equation, allowing us to find values that satisfy the equation when set to zero. Understanding the elements of a quadratic polynomial and how to manipulate them is essential for solving quadratic equations more effectively.

Algebraic Expressions

Algebraic expressions are mathematical phrases that involve numbers, variables, and operations. They form the building blocks of algebra, enabling us to express relationships and solve problems systematically. The polynomial given in our exercise, \(a^2 - 9a + 20\), is an example of an algebraic expression.

• Expressions can include operations such as addition, subtraction, multiplication, and division.
• Variables like \(a\) represent unknown values and play a critical role in algebraic expressions.
• Constant terms, like the number 20 in this expression, represent fixed values.

Understanding how to manipulate algebraic expressions through operations such as factoring, expansion, and simplification is crucial in solving equations and other challenges in algebra. Knowing how each component in the expression interacts allows us to solve not just polynomials, but a wide array of mathematical problems efficiently.