About Adding & Subtracting Polynomials:
In order to add two or more polynomials together, we simply combine like terms. When we need to subtract one polynomial from another, we change the operation into the addition of the opposite. Recall that a - b = a + (-b). To use this concept, we begin by placing the polynomial being subtracted away inside of a set of parentheses. Next, we change the subtraction operation into addition and place a "-1" outside of the parentheses. This "-1" will be distributed to each term inside of the parentheses. Once this is done, we can add the two polynomials together by combining any like terms that are present.
Test Objectives
- Demonstrate the ability to determine if two terms are "like terms"
- Demonstrate the ability to add two or more polynomials together
- Demonstrate the ability to perform subtraction with polynomials
- Demonstrate the ability to write a polynomial in standard form
#1:
Instructions: Simplify each.
$$a)\hspace{.2em}(2x^4 - 5) - (5 + 5x^4)$$
$$b)\hspace{.2em}(x^3 - 3x) + (3x + x^3)$$
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#2:
Instructions: Simplify each.
$$a)\hspace{.2em}(5x^2 - 5x^4) + (3x^4 + 3x^2)$$
$$b)\hspace{.2em}(5x^4 - 3x^3) + (4 + x^3) - (7x^3 - 5x^4)$$
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#3:
Instructions: Simplify each.
$$a)\hspace{.2em}(7x^3 - 6x^4) + (5x^4 - x^3) - (6x^4 + 2x^3)$$
$$b)\hspace{.2em}(3x^2 + 8x^4 + 8) - (7 - x^2 + 8x^4) + (5x^2 - x^4 - 1)$$
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#4:
Instructions: Simplify each.
$$a)\hspace{.2em}(4x^2 - 5x - 6) + (6x^2 - 6x + 3) - (4 + 6x^2 - 3x)$$
$$b)\hspace{.2em}(6x^2 - 6y^4) - (7y^4 - 4x^2) + (4y^4 + 6x^2)$$
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#5:
Instructions: Simplify each.
$$a)\hspace{.2em}(7x^2 - 6x^3) + (y - 7x^2) + (4y + 2x^2)$$
$$b)\hspace{.2em}(y + 2xy + 5x^4y^4) - (6x^4y^4 + 6xy - x^4y^3) - (6x^4y^4 - 8x^4y^3)$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}{-}3x^4 - 10$$
$$b)\hspace{.2em}2x^3$$
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#2:
Solutions:
$$a)\hspace{.2em}{-}2x^4 + 8x^2$$
$$b)\hspace{.2em}10x^4 - 9x^3 + 4$$
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#3:
Solutions:
$$a)\hspace{.2em}{-}7x^4 + 4x^3$$
$$b)\hspace{.2em}{-}x^4 + 9x^2$$
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#4:
Solutions:
$$a)\hspace{.2em}4x^2 - 8x - 7$$
$$b)\hspace{.2em}{-}9y^4 + 16x^2$$
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#5:
Solutions:
$$a)\hspace{.2em}{-}6x^3 + 2x^2 + 5y$$
$$b)\hspace{.2em}{-}7x^4y^4 + 9x^4y^3 - 4xy + y$$
