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To subtract polynomials, you should first recognize that polynomials are algebraic expressions consisting of variables raised to non-negative integer powers, and coefficients.
Subtracting one polynomial from another essentially involves dealing with each like term individually. The process is to align corresponding terms, change the signs of the terms of the polynomial being subtracted, and then combine like terms.
For instance, given two polynomials $P(x) = 4x^2 + 3x – 5$ and $Q(x) = x^2 – 2x + 4$, the expression after subtraction would be $(4x^2 + 3x – 5) – (x^2 – 2x + 4) $.
The next step involves applying the distributive property to eliminate the parentheses and changing the sign of each term of the polynomial being subtracted.
This results in $4x^2 + 3x – 5 – x^2 + 2x – 4$. Ensuring that each term is properly aligned enables us to simplify this expression effectively. The simplified result after combining like terms would be $3x^2 + 5x – 9$, granting us the final subtracted polynomial.
Understanding this process is essential for anyone looking to master algebra and engage with more complex mathematics.
Stay with us to see examples that will elucidate the concept, ensuring you can apply these principles with confidence.
Step-by-Step Subtraction of Polynomials With Samples
Subtracting polynomials involves a sequence of strategic steps to arrive at the correct answer.
Step 1: Write the polynomials in their standard form. Ensure that terms are arranged from highest to lowest power of variables.
Step 2: Align the polynomials horizontally or vertically, ensuring that like terms are in the same columns.
Step 3: Apply the distributive property to the second polynomial by changing every term’s sign within the parentheses, effectively multiplying by $-1$.
Here’s a sample subtraction of two polynomials to illustrate these steps.
Given the polynomials $P(x) = 3x^3 + 5x^2 – x + 6$ and $Q(x) = x^3 – 4x^2 + 3x – 2$, subtract $Q(x)$ from $P(x)$:
$P(x) – Q(x) = 3x^3 + 5x^2 – x + 6 – x^3 – 4x^2 + 3x – 2$
Using Step 3, change the signs of $Q(x)$:
$ = 3x^3 + 5x^2 – x + 6 – x^3 +4x^2 -3x +2$
Now, combine like terms:
$$\begin{align*} &= (3x^3 – x^3) + (5x^2 + 4x^2) + (-x – 3x) + (6 + 2) \ &= 2x^3 + 9x^2 – 4x + 8 \end{align*} $$
This final result, $2x^3 + 9x^2 – 4x + 8$, is the polynomial obtained after subtracting $Q(x)$ from $P(x)$. Notice how the like terms were combined by adding or subtracting their coefficients.
In another example, we align the polynomials vertically:
Subtract $5x^2 – 3x + 7$ from $2x^2 + x – 4$:
$$\begin{array}{r} 2x^2+ x – 4 \ (5x^2 – 3x + 7) \ -3x^2 + 4x – 11 \ \end{array}$$
The negative sign distributes, changing the signs of the second polynomial’s terms, after which the process of combining like terms follows. The answer is $-3x^2 + 4x – 11$. By adhering to the steps systematically, one can simplify polynomial subtractions confidently.
Subtraction Methods and Shortcuts
When students tackle the topic of subtracting polynomials, it is crucial to understand the two primary methods: the horizontal method and the vertical method. Both methods rely on the same fundamental principles but differ in layout.
Horizontal Method
The horizontal method involves writing the polynomials one after the other and then adding polynomials with changed signs.
For example, to subtract $9x^2 + 14x – 16$ from $16x^2 – 9$, one would rewrite the latter polynomial with opposite signs and align them horizontally:
$$(16x^2 – 9) – (9x^2 + 14x – 16) = 16x^2 – 9 – 9x^2 – 14x + 16$$
The next step is to combine like terms:
$$ = (16x^2 – 9x^2) – 14x + (16 + 9) $$
$$ = 7x^2 – 14x + 25 $$
Vertical Method
Conversely, the vertical method involves aligning like terms in columns, similar to traditional subtraction. This method can be particularly efficient for polynomials with many terms. Here’s an example using the same polynomials:
$$ \begin{aligned} &\phantom{-}16x^2 – \phantom{0}9 \ -&\phantom{1}9x^2 + 14x – 16 \ &\phantom{1}7x^2 – 14x + 25 \end{aligned} $$
Shortcuts
A shortcut in the process of subtracting involves immediately changing the signs of the polynomial being subtracted and combining like terms as seen in the first step of either method.
This skill becomes useful in adding polynomials as it simplifies the process by converting a subtraction problem into an addition one.
Learners need to understand both methods as they provide a structured approach to dealing with polynomials. Mastery of these methods enables students to solve polynomial subtraction efficiently and accurately.
Conclusion
Subtracting polynomials involves several systematic steps that are easy to grasp with practice. Once the polynomials are arranged in their standard form, each term of the subtracted polynomial must have its sign changed before like terms are combined.
For example, to subtract $$(6x+8y)-(3x-2y)$$ one would first change the sign of the terms in the second polynomial, giving $$(6x+8y)-(3x-2y)$$ as $$(6x+8y)-(+3x-2y)$$ which simplifies to $$(6x+8y)-3x+2y$$
Following this, one combines like terms, resulting in $$6x – 3x + 8y + 2y$$ which simplifies further to $$(6x-3x) + (8y+2y)$$ ultimately yielding the answer: $$3x + 10y$$
Whether the polynomials are subtracted horizontally or vertically, the process ensures accuracy by keeping terms organized. Diligent calculation and attention to the signs of each term prevent common errors.
Mastery of polynomial subtraction provides a foundation for more advanced algebraic manipulations, following the same principles of like-term management and sign accuracy.
The skills learned through practice with polynomials extend to broader mathematical concepts, highlighting the importance of a strong algebraic foundation.