Textbook Question
Perform the indicated operations. Indicate the degree of the resulting polynomial.
1088
views
0. Review of Algebra
Polynomials Intro
3:21 minutesProblem 87
Textbook Question
In Exercises 83–94, find each product.(5x + 7y − 2)(5x + 7y + 2)
Verified step by step guidance1
Recognize the expression as a product of two binomials in the form \((a - b)(a + b)\), where \(a = 5x + 7y\) and \(b = 2\).
Recall the difference of squares formula: \((a - b)(a + b) = a^2 - b^2\).
Substitute \(a = 5x + 7y\) and \(b = 2\) into the formula: \((5x + 7y)^2 - 2^2\).
Calculate \((5x + 7y)^2\) by expanding it: \((5x + 7y)(5x + 7y)\).
Calculate \(2^2\) to complete the expression: \((5x + 7y)^2 - 4\).
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polynomial Multiplication
Polynomial multiplication involves distributing each term in one polynomial to every term in another polynomial. This process is often facilitated by using the distributive property, which ensures that all combinations of terms are accounted for. In the given expression, each term in the first polynomial must be multiplied by each term in the second polynomial.
Recommended video:
03:42
Finding Zeros & Their Multiplicity
Difference of Squares
The expression (a - b)(a + b) represents the difference of squares, which simplifies to a² - b². This concept is crucial for recognizing patterns in polynomial multiplication, particularly when the polynomials are structured as a sum and difference of the same terms. In the provided question, recognizing this pattern can simplify the multiplication process.
Recommended video:
06:24Solving Quadratic Equations by Completing the Square
Combining Like Terms
After multiplying polynomials, the next step is to combine like terms, which are terms that have the same variable raised to the same power. This process simplifies the expression into its most concise form. Understanding how to identify and combine like terms is essential for arriving at the final answer in polynomial expressions.
Recommended video:
5:22Combinations
Related Videos
Related Practice