6.3: Add and subtract polynomial expressions
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Now that we have discussed exponent rules in great lengths, it is time to discuss polynomials and their operations.
- A monomial in one variable is the product of a coefficient and a variable raised to a positive integer exponent. A monomial is of the form \[ax^k,\nonumber\] where \(a\) is the coefficient, \(x\) is the variable (and base), and \(k\) is the degree of the monomial. Recall, \(k\) is a positive integer.
- A binomial in one variable is the sum of two monomials.
- A polynomial is the sum or difference of monomials. The degree of a polynomial is the highest degree of all the terms in the polynomial.
Rewrite the polynomial in standard form and identify the coefficients, variable terms, and degree of the polynomial \[-12x^2+x^3-x+2\nonumber\]
Solution
The standard form of a polynomial is where the polynomial is written with descending exponents:
\[x^3-12x^2-x+2\nonumber\]
The coefficients are \(1,\: −12,\: −1,\) and \(2\); the variable terms are \(x^3,\: −12x^2,\: −x\). The degree of the polynomial is \(3\) because that is the highest degree of all terms.
Evaluate Polynomial Expressions
If we are given a value for the variable in a polynomial, we can evaluate the polynomial.
Evaluate \(2x^2-4x+6\) when \(x=-4\).
Solution
We plug-n-chug \(x = −4\) for every \(x\) and simplify.
\[\begin{array}{rl}2x^2-4x+6&\text{Plug-n-chug }x=-4 \\ 2(\color{blue}{-4}\color{black}{})^2-4(\color{blue}{-4}\color{black}{})+6&\text{Evaluate} \\ 32+16+6&\text{Simplify} \\ 54&\text{Value of the polynomal when }x=-4\end{array}\nonumber\]
It is important to be careful with negative variables, and exponents. Recall, the exponent is only applied to its base. For example, \(−3^2 = −9\) because we evaluate \(3^2\) first, then multiply by a negative. On the other hand, \((−3)^2 = 9\) because we evaluate the entire base in parenthesis as \(−3\cdot −3 = 9\). In math, if it looks different, then it is different.
Evaluate \(-x^2+2x+6\) when \(x=3\).
Solution
\[\begin{array}{rl}-x^2+2x+6&\text{Plug-n-chug }x=3 \\ -(\color{blue}{3}\color{black}{})^2+2(\color{blue}{3}\color{black}{})+6&\text{Evaluate} \\ -9+6+6&\text{Simplify} \\ 3&\text{Value of the polynomial when }x=3\end{array}\nonumber\]
Ada Lovelace, in 1842, described Charles Babbage’s Difference Engine that would be used to calculate values of polynomials. Her work became the foundation for what would become the modern computer (the programming language Ada was named in her honor) more than 100 years after her death from cancer.
Add and Subtract Polynomial Expressions
Generally, when working with polynomials, we rarely know the value of the variable, so, next, we simplify polynomial expressions by adding and subtracting them. We will combine like terms.
Recall. Like terms are terms with the same variable(s) as the base and exponent.
Add: \((4x^3-2x+8)+(3x^3-9x^2-11)\)
Solution
We can add by combining like terms.
\[\begin{array}{rl}(4x^3-2x+8)+(3x^3-9x^2-11)&\text{Rewrite without parenthesis} \\ 4x^3-2x+8+3x^3-9x^2-11&\text{Add like terms} \\ 7x^3-9x^2-2x-3&\text{Sum}\end{array}\nonumber\]
Be sure to write the sum in standard form.
Subtract: \((5x^2-2x+7)-(3x^2+6x-4)\)
Solution
\[\begin{array}{rl}(5x^2-2x+7)-(3x^2+6x-4)&\text{Rewrite without parenthesis} \\ 5x^2-2x+7\color{blue}{-}\color{black}{}3x^2\color{blue}{-}\color{black}{}6x\color{blue}{+}\color{black}{}4&\text{Subtract like terms} \\ 2x^2-8x+11&\text{Difference}\end{array}\nonumber\]
Simplify: \((2x^2 − 4x + 3) + (5x^2 − 6x + 1) − (x^2 − 9x + 8)\)
Solution
\[\begin{array}{rl}(2x^2-4x+3)+(5x^2-6x+1)-(x^2-9x+8)&\text{Rewrite without parenthesis} \\ 2x^2-4x+3+5x^2-6x+1\color{blue}{-}\color{black}{}x^2\color{blue}{+}\color{black}{}9x\color{blue}{-}\color{black}{}8&\text{Combine like terms} \\ 6x^2-x-4&\text{Simplified expression}\end{array}\nonumber\]
Add and Subtract Polynomial Expressions Homework
Evaluate the expression for the given value.
\(−a^3 − a^2 + 6a − 21\) when \(a = −4\)
\(n^2 + 3n − 11\) when \(n = −6\)
\(n^3 − 7n^2 + 15n − 20\) when \(n = 2\)
\(n^3 − 9n^2 + 23n − 21\) when \(n = 5\)
\(−5n^4 − 11n^3 − 9n^2 − n − 5\) when \(n = −1\)
\(x^4 − 5x^3 − x + 13\) when \(x = 5\)
\(x^2 + 9x + 23\) when \(x = −3\)
\(−6x^3 + 41x^2 − 32x + 11\) when \(x = 6\)
\(x^4 − 6x^3 + x^2 − 24\) when \(x = 6\)
\(m^4 + 8m^3 + 14m^2 + 13m + 5\) when \(m = −6\)
Simplify. Write the answer in standard form.
\((5p − 5p^4) − (8p − 8p^4)\)
\((7m^2 + 5m^3) − (6m^3 − 5m^2)\)
\((3n^2 + n^3) − (2n^3 − 7n^2)\)
\((x^2 + 5x^3) + (7x^2 + 3x^3)\)
\((8n + n^4) − (3n − 4n^4)\)
\((3v^4 + 1) + (5 − v^4)\)
\((1 + 5p^3) − (1 − 8p^3)\)
\((6x^3 + 5x) − (8x + 6x^3)\)
\((5n^4 + 6n^3) + (8 − 3n^3 − 5n^4)\)
\((8x^2 + 1) − (6 − x^2 − x^4)\)
\((3 + b^4) + (7 + 2b + b^4)\)
\((1 + 6r^2) + (6r^2 − 2 − 3r^4)\)
\((8x^3 + 1) − (5x^4 − 6x^3 + 2)\)
\((4n^4 + 6) − (4n − 1 − n^4)\)
\((2a + 2a^4) − (3a^2 − 5a^4 + 4a)\)
\((6v + 8v^3) + (3 + 4v^3 − 3v)\)
\((4p^2 − 3 − 2p) − (3p^2 − 6p + 3)\)
\((7 + 4m + 8m^4) − (5m^4 + 1 + 6m)\)
\((4b^3 + 7b^2 − 3) + (8 + 5b^2 + b^3)\)
\((7n + 1 − 8n^4) − (3n + 7n^4 + 7)\)
\((3 + 2n^2 + 4n^4 ) + (n^3 − 7n^2 − 4n^4)\)
\((7x^2 + 2x^4 + 7x^3) + (6x^3 − 8x^4 − 7x^2)\)
\((n − 5n^4 + 7) + (n^2 − 7n^4 − n)\)
\((8x^2 + 2x^4 + 7x^3 ) + (7x^4 − 7x^3 + 2x^2)\)
\((8r^4 − 5r^3 + 5r^2 ) + (2r^2 + 2r^3 − 7r^4 + 1)\)
\((4x^3 + x − 7x^2) + (x^2 − 8 + 2x + 6x^3)\)
\((2n^2 + 7n^4 − 2) + (2 + 2n^3 + 4n^2 + 2n^4)\)
\((7b^3 − 4b + 4b^4) − (8b^3 − 4b^2 + 2b^4 − 8b)\)
\((8 − b + 7b^3) − (3b^4 + 7b − 8 + 7b^2) + (3 − 3b + 6b^3)\)
\((1 − 3n^4 − 8n^3) + (7n^4 + 2 − 6n^2 + 3n^3) + (4n^3 + 8n^4 + 7)\)
\((8x^4 + 2x^3 + 2x) + (2x + 2 − 2x^3 − x^4) − (x^3 + 5x^4 + 8x)\)
\((6x − 5x^4 − 4x^2) − (2x − 7x^2 − 4x^4 − 8) − (8 − 6x^2 − 4x^4)\)