Problem 69 - Algebra A Combined Function

Question

Multiply. \((a+4)\left(a^{2}-6 a+6\right)\)

Step-by-Step Solution

Answer

The product is \(a^3 - 2a^2 - 18a + 24\).

Step 1: Distribute the First Term

Use the distributive property to multiply the first term of \((a + 4)\), which is \(a\), by each term in \((a^2 - 6a + 6)\). This gives us:\(a \times (a^2 - 6a + 6) = a \cdot a^2 - a \cdot 6a + a \cdot 6.\) Which simplifies to:\[a^3 - 6a^2 + 6a.\]

Step 2: Distribute the Second Term

Next, distribute the second term of \((a + 4)\), which is \(4\), by each term in \((a^2 - 6a + 6)\). This gives us:\( 4 \times (a^2 - 6a + 6) = 4 \cdot a^2 - 4 \cdot 6a + 4 \cdot 6.\)Which simplifies to:\[4a^2 - 24a + 24.\]

Step 3: Combine Like Terms

Add the two expressions obtained from Step 1 and Step 2 together:\(a^3 - 6a^2 + 6a + 4a^2 - 24a + 24.\)Combine like terms:\[a^3 - 2a^2 - 18a + 24.\]

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that helps us multiply a single term by a polynomial, essentially allowing us to "distribute" the multiplication over addition or subtraction inside parentheses. This is expressed through the formula: \[ a(b + c) = ab + ac \] In the exercise, we applied the distributive property by multiplying each term of the binomial \((a + 4)\) by each term of the trinomial \((a^2 - 6a + 6)\).

• First, the term \(a\) from \((a + 4)\) was multiplied by \(a^2\), \(-6a\), and \(6\).
• Second, the term \(4\) was similarly distributed across the trinomial.

This step-by-step use of the distributive property results in new terms which we can then work on simplifying further.

Combining Like Terms

After applying the distributive property, the next step involves combining like terms to simplify the expression. "Like terms" are terms that have the exact same variables raised to the same power. For example, \(3a^2\) and \(-6a^2\) are like terms because they both contain \(a^2\). In our exercise, after distributing, we ended up with the expression:\[ a^3 - 6a^2 + 6a + 4a^2 - 24a + 24. \]Here, like terms can be grouped together:

• Combine \(-6a^2\) and \(4a^2\) to get \(-2a^2\).
• Combine \(6a\) and \(-24a\) to get \(-18a\).

This simplifies our expression to: \[ a^3 - 2a^2 - 18a + 24 \] Combining like terms makes our polynomial easier to interpret and further work with.

Monomials

Monomials are algebraic expressions consisting of only a single term, which can be just a number (a constant), a variable, or a product of numbers and variables. Each monomial is an independent unit in expressions like polynomials. Some examples include:

• \(7\)
• \(x\)
• \(-3y^2\)

In the exercise, when distributing and expanding the expression, we work with several monomials like \(a^3\), \(-6a^2\), and \(4\). Carefully managing these monomials and ensuring they are correctly combined in later steps is crucial for accurate problem-solving.

Quadratic Expression

A quadratic expression is a polynomial of degree 2, typically in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). In the context of expanding and simplifying polynomials, quadratic expressions often show up when multiplying two binomials or certain terms during polynomial multiplication. Our original expression began as a mixture of quadratic and linear components within the trinomial \((a^2-6a+6)\).

• The term \(a^2\) indicates its quadratic nature, initially defined by the highest degree term.
• As we distribute and combine, our resultant expression \(a^3 - 2a^2 - 18a + 24\) adds a cubic component, a clear progression from a quadratic starting point into a more complex polynomial.

Understanding how quadratic expressions shift and transform during multiplication helps grasp broader polynomial dynamics.