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# Polynomial Division

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it is a project to teach people how to do a long division math involving polynomials.

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### Polynomial Division

1. 1. Dividing the same variable when you divide two terms with a common variable. You subtract the exponent of the division. Ex) x ³ x² = x³-² = x Ex) (2x³ + 4x² + 6x) 2x = 2x³ + 4x² + 6x 2x 2x 2x = 2x² + 2x + 3 Anything to the power of zero is always one
2. 2. Simplification and Reduction <ul><li>Simplify: x ² + 9x + 14 </li></ul><ul><li>x + 7 </li></ul><ul><li>This kind a question can be done in either two ways: you can factor the quadratic and then cancel out the common factor, like this. </li></ul><ul><li>x² + 9x + 14 </li></ul><ul><li>x + 7 </li></ul><ul><li>= (x + 2) (x + 7) </li></ul><ul><li>x + 7 </li></ul><ul><li>= x + 2 </li></ul><ul><li>You can also solve for this problem by doing long division. Like this </li></ul><ul><li>x + 2 </li></ul><ul><li>x + 7 x² + 9x + 14 </li></ul><ul><li>- ( x² + 7x ) </li></ul><ul><li>= 0 + 2x + 14 </li></ul><ul><li>- ( 2x + 14 ) </li></ul><ul><li>0 + 0 </li></ul><ul><li>In this case we have no reminder, so the answer is just x+ 2 </li></ul>
3. 3. Long Division <ul><li>4x ² - 2x +3 quotient </li></ul><ul><li>X + 1 4x ³ + 2x² + x + 5 dividend </li></ul><ul><li>- (4x ³ + 4x² ) </li></ul><ul><li>= 0 - 2x ² + x </li></ul><ul><li>- (-2x² - 2x) </li></ul><ul><li>0 + 3x + 5 </li></ul><ul><li>- (3x + 3) </li></ul><ul><li>= 2 this is called the remainder </li></ul>Divisor
4. 4. Dividing polynomial with “missing terms” <ul><li>When dividing a polynomial with a missing term, you consider that missing term as a zero. It doesn’t matter if you put a negative sign or positive sign, it is the same. </li></ul><ul><li>ex) 8x ³ + 1 </li></ul><ul><li>2x+ 1 </li></ul><ul><li>4x² - 2x + 1 </li></ul><ul><li>2x + 1 8x³ + 0x² + 0x + 1 </li></ul><ul><li>- (8x³ + 4x²) </li></ul><ul><li>= 0 -4x² + 0x </li></ul><ul><li>- ( -4x² - 2x) </li></ul><ul><li>= 0 + 2x + 1 </li></ul><ul><li>- (2x + 1) </li></ul><ul><li>0+ 0 </li></ul><ul><li>In this question there is no remainder </li></ul>
5. 5. Expression for polynomial “A” <ul><li>Ex) A ( x+ 1) – ( 2x + x + 14) = (3x – 2) (x + 4) </li></ul><ul><li>A (x + 1) = (3x – 2) (x+ 4) + (2x + x + 14) </li></ul><ul><li>= (3x ² + 12x -2x – 8)+ (2x + x+ 14) </li></ul><ul><li>A (x + 1) = (3x² + 13x + 6) </li></ul><ul><li>x + 1 (x + 1) </li></ul><ul><li>3x + 10 R= -4 </li></ul><ul><li>x + 1 3x² + 13x + 6 </li></ul><ul><li>- (3x² + 3x) </li></ul><ul><li>= 0 + 10x + 6 </li></ul><ul><li>- (10x + 10) </li></ul><ul><li>= - 4 </li></ul>
6. 6. Problem solving <ul><li>A B </li></ul><ul><li>D C </li></ul><ul><li>F E </li></ul>A1= 3x ² - 4x - 7 A2= 4x ² - 3x - 7 x + 1 x + 1 Find DE Find AB
7. 7. The diagram above <ul><li>A1= 3x² - 4x – 7 </li></ul><ul><li>A2= 4x² - 3x – 7 </li></ul><ul><li>BC= x + 1 </li></ul><ul><li>FE= x + 1 </li></ul><ul><li>Find AF, DE, DC, and AB </li></ul><ul><li>To find AF you need to do a long division by using area one and FE </li></ul><ul><li>3x – 7 </li></ul><ul><li>x + 1 3x² - 4x – 7 </li></ul><ul><li>- (3x² + 3x) </li></ul><ul><li>= -7x -7 </li></ul><ul><li>+ ( -7x – 7) </li></ul><ul><li>= o </li></ul><ul><li>so AF= 3x - 7 </li></ul>
8. 8. The same diagram <ul><li>AF= DE + BC </li></ul><ul><li>so since we have AF and BC we can find DE by subtracting AF by BC and that gives us DE </li></ul><ul><li>(3x – 7) – x - 1 = DE </li></ul><ul><li>DE= 2x – 8 </li></ul><ul><li>To find DC you need to use area two divide by BC </li></ul><ul><li>4x – 7 </li></ul><ul><li>x + 1 4x² - 3x – 7 </li></ul><ul><li>- (4x² + 4x) </li></ul><ul><li>= 0 -7x – 7 </li></ul><ul><li>+ (-7x – 7) </li></ul><ul><li>= 0 </li></ul><ul><li>DC= 4x - 7 </li></ul>
9. 9. The same diagram <ul><li>now we have FE and DC and we are asked to find AB. </li></ul><ul><li>What you do is that you take FE + DC and that gives you AB </li></ul><ul><li>AB = 4x – 7 + x + 1 </li></ul><ul><li>= 5x - 6 </li></ul>
10. 10. Solving for a triangle by using Long Division <ul><li>A= 14x ² + x – 3 </li></ul><ul><li>B= 4x + 2 </li></ul><ul><li>Find the Height </li></ul><ul><li>To solve for this kind a question you need to know that A= B * H </li></ul><ul><li>2 </li></ul><ul><li>So 2A= B* H </li></ul><ul><li>So now you substitute the numbers in </li></ul><ul><li>2 ( 14x ² + x – 3) = (4x + 2) H </li></ul><ul><li>28x² + 2x – 6 = (4x + 2) H </li></ul><ul><li>28x² + 2x – 6 = H </li></ul><ul><li>4x + 2 </li></ul>
11. 11. the same triangle <ul><li>Now to find the height you need to do long division, like this </li></ul><ul><li>28x² + 2x – 6 </li></ul><ul><li>4x + 2 </li></ul><ul><li>7x – 3 </li></ul><ul><li>4x + 2 28x² + 2x – 6 </li></ul><ul><li>- ( 28x² + 14x) </li></ul><ul><li>= 0 - 12x – 6 </li></ul><ul><li>+ (-12x – 6) </li></ul><ul><li>= 0 </li></ul><ul><li>the height = 7x - 3 </li></ul>
12. 12. Solving a Rectangle by using long division <ul><li>Here are the given information </li></ul><ul><li>A= 8x ² + 22x + 15 </li></ul><ul><li>H= 2x + 3 </li></ul><ul><li>Find the base of this rectangle </li></ul><ul><li>So to find the base of a rectangle you need to know two things, and those two things are area and the Height. </li></ul><ul><li>So since we have given those two piece of information we can find out what the base is by doing long division. </li></ul><ul><li>4x + 5 </li></ul><ul><li>2x + 3 8x ² + 22x + 15 </li></ul><ul><li>- ( 8x ² + 12x) </li></ul><ul><li>= 10 + 15 </li></ul><ul><li>- (10x + 15) </li></ul><ul><li>= 0 + 0 </li></ul>