Question

Use the method of your choice to factor the polynomial completely. Explain your reasoning. $$ a^6+a^5-30 a^4 $$

Short answer

The factorized form of the polynomial \( a^6 + a^5 -30a^4 \) is \( a^4 (a - 5)(a + 6) \).

Step-by-step solution

Identifying Common Factors

Looking at the polynomial \( a^6 + a^5 -30a^4 \), one common factor to observe in each term is \( a^4 \). This is identified by looking for the lowest common exponent of 'a' across all terms, which is 4.

Factoring Out Common Factors

We can factor out \( a^4 \) from the polynomial, which then becomes \( a^4 ( a^2 + a - 30 ) \).

Factoring Quadratic Expression

The quadratic expression \( a^2 + a - 30 \) can still be factored. The goal here is to find two numbers that add up to 1 (the coefficient in front of 'a') and multiply to -30 (the constant). Those numbers are 6 and -5. Thus, \( a^2 + a - 30 \) factorizes to \( (a - 5)(a + 6) \).

Writing The Final Factorized Polynomial

Substitute \( (a - 5)(a + 6) \) back into the polynomial to get the final fully factorized polynomial: \( a^4 (a - 5)(a + 6) \).

Key concepts

Common Factors

When factoring polynomials, identifying common factors is a crucial first step. A common factor is an expression that can be evenly divided out of each term in the polynomial. In our exercise, the polynomial can be written as a^6 + a^5 - 30 a^4. By inspecting each term, we notice that they all share a^4 as a factor.

This commonality allows us to 'pull out' a^4, which means dividing each term by a^4 and placing it in front of a set of parentheses. This process simplifies the polynomial and sets the stage for further factoring. It's similar to how you would take ‘4’ out of the numbers 8, 12, and 20; they all can be divided by 4, which simplifies the numbers to 2, 3, and 5, respectively.

The concept of common factors not only simplifies the original polynomial but it also makes the remaining algebraic expression more manageable for further operations, such as factoring quadratic expressions.

Factoring Quadratic Expressions

As we proceed with factoring, we face a quadratic expression like a^2 + a - 30. A quadratic expression is characterized by the highest degree of the variable being two (i.e., a^2). Our goal here is to break down this part of the polynomial into two binomials.

To do this effectively, one must look for two numbers that multiply to give the constant term (-30 in this case) and add up to the coefficient of the linear term (1 in this case). In our example, the numbers 6 and -5 do just that: 6 * -5 = -30 and 6 + (-5) = 1.

These two values are then used to create the binomial factors (a - 5) and (a + 6), leading to a factored quadratic expression. Factoring quadratics is a vital skill in algebra that helps transform complex polynomials into more straightforward, solvable equations.

Algebraic Expressions

Algebraic expressions, like the ones we see in our polynomial a^6 + a^5 - 30a^4, consist of numbers, variables, and operators. They convey a wide range of mathematical relationships. In solving algebra problems, understanding how to manipulate these expressions is key.

Techniques such as combining like terms, which are terms with the same variables raised to the same powers, or distributing factors across terms are essential. In factoring, algebraic expressions are altered in such a way that their product equates to the original polynomial.

The successful factoring of a^4 (a^2 + a - 30) into a^4 (a - 5)(a + 6) showcases this manipulation. Each factor is an algebraic expression itself that, when multiplied together, returns the initial, more complex expression, providing a clearer view of potential roots or solutions.