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1. Polynomials & Factoring
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Integrated Math 2

1.01 Adding and subtracting polynomials

Lesson
Practice
Lesson

Concept summary

In the same way that we can add and subtract algebraic expressions, we can also add and subtract polynomials.

When adding or subtracting polynomials, we add or subtract the whole expressions and simplify by combining any like terms.

  • To find the sum/difference of two polynomials using a horizontal algorithm:

    1. Group each polynomial inside a pair of parentheses and add/subtract one from the other.
    2. If we are subtracting one polynomial from the other, we then distribute the subtraction and remove any parentheses.-(ax^2+bx+c)=-ax^2-bx-c
    3. Combine any like terms. We can use the commutative and associative property of addition (for both positive and negative terms) to group the like terms beforehand.

  • To find the sum or difference of two polynomials using a vertical algorithm:

    1. Align the like terms in a vertical algorithm with the appropriate operation.
    2. Apply the operation to each column of the algorithm. For polynomials, there is no carry over or borrowing between columns.

The sum or difference of any two polynomials will always be a polynomial.

Worked examples

Example 1

Simplify the expression:\left(3x^2-5x+1\right)-\left(x^2+7x-10\right)

Solution

\displaystyle \left(3x^2-5x+1\right)-\left(x^2+7x-10\right)\displaystyle =\displaystyle 3x^2-5x+1-x^2-7x+10Distribute the subtraction
\displaystyle =\displaystyle \left(3x^2-x^2\right)+\left(-5x-7x\right)+\left(1+10\right)Group the like terms together
\displaystyle =\displaystyle 2x^2-12x+11Simplify

Example 2

Consider the polynomials x^3-6x+2 and x^2+9x+7.

a

Find the sum of the two polynomials.

Approach

Since we want to find the sum of the two polynomials, we want to add them together.

Solution

\displaystyle \text{Sum}\displaystyle =\displaystyle \left(x^3-6x+2\right)+\left(x^2+9x+7\right)Add the polynomials together
\displaystyle =\displaystyle x^3-6x+2+x^2+9x+7Remove the parentheses
\displaystyle =\displaystyle x^3+x^2+3x+9Combine the like terms

Reflection

If we want to use the vertical algorithm method, we need to make sure we correctly align the like terms.

\begin{aligned} & & x^3 & & & & - & 6x & + 2 \\ + & & & & & x^2 & + & 9x & + 7 \\ \hline \\ & & x^3 & & + & x^2 & + & 3x & + 9 \end{aligned}

b

Explain why the sum of two polynomials is also a polynomial.

Solution

A polynomial is a collection of terms in the form mx^n where m is a real number and n is a non-negative integer.

When adding polynomials together, terms are either combined if they have the same value of n or left alone if not. Therefore, we can see that the result will always be a collection of terms in the form mx^n, because there is no way to introduce terms that are not of this form. This means that the sum is also a polynomial.

Reflection

We can use the same explanation for why the difference of two polynomials is also a polynomial, and we can extend this explanation to include the sum or difference of any number of polynomials.

Outcomes

M2.A.APR.A.1

Add, subtract, and multiply polynomials. Use these operations to demonstrate that polynomials form a closed system that adhere to the same properties of operations as the integers.

M2.MP1

Make sense of problems and persevere in solving them.

M2.MP5

Use appropriate tools strategically.

M2.MP6

Attend to precision.

M2.MP7

Look for and make use of structure.

M2.MP8

Look for and express regularity in repeated reasoning.

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