About Special Polynomial Products:
When we learn how to multiply polynomials, we often encounter special cases that occur very often. These special cases involve multiplying two or more binomials. Instead of grinding out the work each time, we can memorize a formula in each case, and then apply the formula to our given problem. These types of problems are known as special products or special polynomial products.
Test Objectives
- Demonstrate an understanding of the rules of exponents
- Demonstrate the ability to multiply conjugates
- Demonstrate the ability to square a binomial
- Demonstrate the ability to cube a binomial
#1:
Instructions: Find each product.
$$a)\hspace{.2em}(3x + 7)(3x - 7)$$
$$b)\hspace{.2em}(5x - 6)(5x + 6)$$
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#2:
Instructions: Find each product.
$$a)\hspace{.2em}(6x - 1)^2$$
$$b)\hspace{.2em}(x + 6)^2$$
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#3:
Instructions: Find each product.
$$a)\hspace{.2em}(7 + 6x)(7 - 6x)$$
$$b)\hspace{.2em}(2x^2 + 3)^2$$
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#4:
Instructions: Find each product.
$$a)\hspace{.2em}(8x - 2)^3$$
$$b)\hspace{.2em}(5x + 1)^3$$
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#5:
Instructions: Find each product.
$$a)\hspace{.2em}(3x^2 - y^2)^3$$
$$b)\hspace{.2em}(5x^2 + 3y^4)^3$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}9x^2 - 49$$
$$b)\hspace{.2em}25x^2 - 36$$
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#2:
Solutions:
$$a)\hspace{.2em}36x^2 - 12x + 1$$
$$b)\hspace{.2em}x^2 + 12x + 36$$
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#3:
Solutions:
$$a)\hspace{.2em}{-}36x^2 + 49$$
$$b)\hspace{.2em}4x^4 + 12x^2 + 9$$
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#4:
Solutions:
$$a)\hspace{.2em}512x^3 - 384x^2 + 96x - 8$$
$$b)\hspace{.2em}125x^3 + 75x^2 + 15x + 1$$
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#5:
Solutions:
$$a)\hspace{.2em}27x^6 - 27x^4y^2 + 9x^2y^4 - y^6$$
$$b)\hspace{.2em}27y^{12}+ 135x^2y^8 + 225x^4y^4 + 125x^6$$
