Problem 7 - Precalculus

Question

Carry out the indicated operations. (a) \((T+3)(T-2)\) (b) \((\tan \theta+3)(\tan \theta-2)\)

Step-by-Step Solution

Answer

(a) \(T^2 + T - 6\), (b) \((\tan \theta)^2 + \tan \theta - 6\)."

Step 1: Apply the Distributive Property

To solve part (a) \((T+3)(T-2)\), apply the distributive property (also known as FOIL for binomials):\((T+3)(T-2) = T(T-2) + 3(T-2)\).This means you distribute each term in the first parentheses across all terms in the second parentheses.

Step 2: Distribute Each Term

Distribute each term within both binomials individually:For the term \(T(T-2)\), distribute the \(T\):\(T \cdot (T-2) = T^2 - 2T\).Then, for the term \(3(T-2)\), distribute the \(3\):\(3 \cdot (T-2) = 3T - 6\).

Step 3: Combine Like Terms

Now combine all the terms from the distribution in Step 2:Combine:\(T^2 - 2T + 3T - 6\).Simplify by combining the like terms \(-2T + 3T\), which gives \(T^2 + T - 6\).

Step 4: Apply the Distributive Property for Part (b)

To solve part (b)\((\tan \theta+3)(\tan \theta-2)\), apply the distributive property:\((\tan \theta+3)(\tan \theta-2) = \tan \theta(\tan \theta-2) + 3(\tan \theta-2)\).

Step 5: Distribute Each Term for Part (b)

Distribute each term for part (b) individually:For \(\tan \theta(\tan \theta-2)\), distribute the \(\tan \theta\):\((\tan \theta) \cdot (\tan \theta-2) = (\tan \theta)^2 - 2\tan \theta\).Then, for \(3(\tan \theta-2)\), distribute the \(3\):\(3 \cdot (\tan \theta-2) = 3\tan \theta - 6\).

Step 6: Combine Like Terms for Part (b)

Now, combine all the terms for part (b) from Step 5:Combine:\((\tan \theta)^2 - 2\tan \theta + 3\tan \theta - 6\).Simplify by combining the like terms \(-2\tan \theta + 3\tan \theta\), which gives \((\tan \theta)^2 + \tan \theta - 6\).

Key Concepts

Binomials

A binomial is a polynomial expression that contains exactly two terms. It often takes the form of either

• two variables, like \(x + y\), or
• a variable and a number, like \(x - 3\).

Understanding binomials is crucial because they frequently appear in algebraic expressions and equations. Each term in a binomial is separated by an addition or subtraction operator. This forms a small but powerful unit in algebra, which helps in simplifying complex calculations.
Binomials are the building blocks for polynomial expressions, and mastering them allows you to tackle more complicated algebra problems.

• When multiplying binomials, you employ the distributive property.
• Each term in the first binomial multiplies each term in the second binomial.

Combining Like Terms

Combining like terms is a fundamental technique in simplifying algebraic expressions. This process reduces expressions by merging terms that contain the same variables. For example, in the expression \(T^2 - 2T + 3T - 6\), notice the terms \(-2T\) and \(3T\)\; these are like terms because they both contain the variable \(T\) raised to the same power.
To combine them, add or subtract the coefficients: \(-2+3 = 1\). Thus, the expression simplifies to \(T^2 + T - 6\).
By combining like terms, you can rearrange and simplify larger algebraic expressions into simpler forms.

• Look for terms that share the same variable and exponent.
• Only these terms are eligible for combining.
• Combine by adding or subtracting their coefficients while keeping the variable part unchanged.

Polynomial Multiplication

Polynomial multiplication is an essential skill in algebra, particularly when working with larger expressions. This technique extends on simple binomial multiplication, where multiple terms are involved. In this case, using the distributive property is key.
To multiply polynomials such as \(T+3\) and \(T-2\), each term in one polynomial is multiplied by each term in the other. This method can be remembered using the FOIL method, which stands for:

• First: Multiply the first terms in each binomial (e.g., \(T imes T = T^2\)).
• Outer: Multiply the outer border terms (e.g., \(T imes -2 = -2T\)).
• Inner: Multiply the inner border terms (e.g., \(3 imes T = 3T\)).
• Last: Multiply the last terms in each binomial (e.g., \(3 imes -2 = -6\)).

After foil application, combine the like terms using addition or subtraction of coefficients to arrive at the final simplified expression. The product \(T^2 + T - 6\) comes from executing these steps on binomials. This method is universal and applicable to polynomial multiplication of any size.