Problem 66 - Elementary and Intermediate Algebra

Question

Subtract \(\left(32 x^{2}-17 x+45\right)\) from the sum of \(\left(23 x^{2}-12 x-7\right)\) and \(\left(-11 x^{2}+12 x+7\right)\)

Step-by-Step Solution

Answer

The result is \\-20x^{2} + 17x - 45\\.

Step 1: Find the Sum of Two Expressions

First, we find the sum of the expressions \(23x^{2} - 12x - 7\) and \(-11x^{2} + 12x + 7\). To do this, add the corresponding coefficients. \[ (23x^{2} - 12x - 7) + (-11x^{2} + 12x + 7) \] \[ (23x^{2} - 11x^{2}) + (-12x + 12x) + (-7 + 7) = 12x^{2} + 0x + 0 \] So, the sum is \12x^{2}\.

Step 2: Subtract the Given Expression

Now, subtract the expression \(32x^{2} - 17x + 45\) from the sum obtained in Step 1, which is \(12x^{2}\). \[ 12x^{2} - (32x^{2} - 17x + 45) \] Distribute the negative sign: \[ 12x^{2} - 32x^{2} + 17x - 45 \] Subtract the corresponding coefficients: \[ (12x^{2} - 32x^{2}) + 17x - 45 = -20x^{2} + 17x - 45 \] Thus, the result of the subtraction is \-20x^{2} + 17x - 45\.

Key Concepts

Polynomial Addition

Polynomial addition is a method where you add two or more polynomials together. It is similar to adding regular numbers but involves adding each corresponding term according to its degree.

• Identify Like Terms: When adding polynomials, focus on "like terms." These terms have the same variable and the same exponent. For example, in the original exercise, both have terms like \(23x^2\) and \(-11x^2\), making them like terms.
• Add the Coefficients: Simply add the coefficients of these like terms and keep the common variable part. In the expression \(23x^2 - 12x - 7\) plus \(-11x^2 + 12x + 7\), the addition steps are to combine \(23x^2\) and \(-11x^2\), \(-12x\) and \(12x\), and \(-7\) and \(7\).
• Combine the Results: After adding each pair of coefficients, write down the polynomial resulting from those combined terms. Using the exercise example, the steps show: \((23 - 11)x^2 + (-12 + 12)x + (-7 + 7)\) resulting in \(12x^2 + 0x + 0\).

This method simplifies the process of understanding and performing polynomial addition step-by-step.

Algebraic Expressions

Algebraic expressions are mathematical statements involving variables, numbers, and operations (such as addition and subtraction). They form the foundational building blocks in algebra.

• Variables and Constants: In an algebraic expression, variables are symbols (like \(x\) or \(y\)) that represent numbers that can change. Constants are fixed numbers, such as 45 in the expression \(32x^2 - 17x + 45\).
• Terms and Operations: An algebraic expression comprises terms, which are http://products of numbers and variables. For example, \(32x^2\) and \(17x\) are terms in \(32x^2 - 17x + 45\). These terms are connected by operations such as addition and subtraction.
• Expression Simplification: The goal is often to simplify expressions by combining like terms. For example, adding the expressions \(23x^2 - 12x - 7\) and \(-11x^2 + 12x + 7\) simplifies to \(12x^2\).

Understanding the structure and elements of algebraic expressions helps in both the simplification and manipulation of polynomials.

Coefficients

Coefficients are key components in polynomials and algebraic expressions, representing the numerical factor of terms.

• Definition: In a term like \(23x^2\), 23 is the coefficient. It shows how many instances of the base (or term) you're considering.
• Role in Polynomial Operations: When adding or subtracting polynomials, focus on the coefficients of like terms. These are the parts you add or subtract, as seen in the expression \(23x^2 - 11x^2\), where you only work with 23 and -11.
• Keeping Track of Signs: Pay attention to the signs in front of coefficients. They affect the outcome, such as when subtracting polynomials, ensuring the negative sign is appropriately distributed: \(12x^2 - 32x^2 + 17x - 45\).

Mastering coefficients accelerates understanding and handling various operations within polynomial expressions effectively.