Polynomials - Revision history

Iwiseman: /* Factoring by Grouping */

2026-04-21T01:11:08Z

Factoring by Grouping

← Older revision		Revision as of 01:11, 21 April 2026
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	~~=Factoring by Grouping=~~
	~~Here we divide the polynomial into two group and take out the GCF~~		addasq
	~~x³ - 4x² + 3x -12~~
	~~So group one~~
	~~x³ - 4x² = x²(x-4)~~
	~~Group two~~
	~~3x -12 = 3(x - 4)~~
	~~So they both have (x-4) so we can say~~
	~~(x-4)(x² + 3)~~
	~~Next example~~
	~~2r³ -6r² + 5r - 15~~
	~~= 2r²(r -3) +5(r -3)~~
	~~= (r-3)(2r² +5)~~
	=Perfect Square Trinomial=		=Perfect Square Trinomial=
	==Intro==		==Intro==

Iwiseman: /* Factoring by Trinomials */

2026-04-21T01:03:32Z

Factoring by Trinomials

@@ Line 12: / Line 12: @@
   = 2r²(r -3) +5(r -3)
   = (r-3)(2r² +5)
 =Perfect Square Trinomial=
 ==Intro==

Iwiseman: /* Greatest Common Factor (GCF) */

2026-04-21T01:00:26Z

Greatest Common Factor (GCF)

← Older revision		Revision as of 01:00, 21 April 2026
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	~~=Greatest Common Factor (GCF)=~~
	~~This is where you look at a polynomial and find, you guest it, the greatest common factor so given~~
	~~3x + 15~~
	~~It would be 3 because~~
	~~3(x + 5)~~
	~~And~~
	~~4x² + 8x = 4x(x+2) GCF = 4x~~
	~~5x² -15x³ = 5x²(1 -3x) GCF = 5x²~~
	=Factoring by Grouping=		=Factoring by Grouping=
	Here we divide the polynomial into two group and take out the GCF		Here we divide the polynomial into two group and take out the GCF

Iwiseman: /* Long Division */

2026-04-21T00:54:48Z

Long Division

← Older revision		Revision as of 00:54, 21 April 2026
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	~~=Long Division=~~
	~~==Example 1==~~
	~~So getting more tricky below we need to follow a process. The question is~~
	~~(x² + 5x + 6) / (x + 2)~~
	~~So we put the denominator on the left like normal long division~~

	~~-----------------------------~~
	~~(x + 2) \| x² + 5x + 6~~

	~~Step 1: Next we divide (x² + 5x + 6) by the first term only~~
	x
	~~-----------------------------~~
	~~(x + 2) \| x² + 5x + 6~~

	~~Step 2: Next we need to multiply everything in the denominator by the result (top row) x and reverse the sign~~
	x
	~~-----------------------------~~
	~~(x + 2) \| x² + 5x + 6~~
	~~-(x² - 2x)~~

	~~Step 3: Now calculate the remainder by take one from the other~~
	x
	~~-----------------------------~~
	~~(x + 2) \| x² + 5x + 6~~
	~~-(x² - 2x)~~
	~~-----------------------------~~
	~~0 + 3x + 6~~

	~~No next we do steps 1-3 again,~~

	~~Step 1: So x / 3x = + 3~~
	~~x + 3~~
	~~-----------------------------~~
	~~(x + 2) \| x² + 5x + 6~~
	~~-(x² - 2x)~~
	~~-----------------------------~~
	~~3x + 6~~

	~~Step 2: Next we need to multiply everything in the denominator by the result (top row) x and reverse the sign~~
	~~x + 3~~
	~~-----------------------------~~
	~~(x + 2) \| x² + 5x + 6~~
	~~-(x² - 2x)~~
	~~-----------------------------~~
	~~3x + 6~~
	~~-(3x +6)~~
	~~-------~~
	0

	~~Step 3: Now calculate the remainder by take one from the other give 0~~

	~~So we can say~~

	~~(x² + 5x + 6) / (x + 2) = x+3~~
	~~==Example 2==~~
	~~So here is the next and shows you what to do when you have something left over. Here is the question and workings~~

	~~2x³+8x²-6x+10 / x-2~~

	~~2x²+12x+18+ 46/(x-2)~~
	~~-----------------------------~~
	~~x-2 \| 2x³+8x²-6x+10~~
	~~-(2x³-4x²)~~
	~~---------------~~
	~~12x²-6x+10~~
	~~-(12x²-24x)~~
	~~-----------~~
	~~18x+10~~
	~~-(18x-36)~~
	~~---------~~
	46
	~~So the answer is 2x²+12x+18 + 46 / x-2 left over.~~

	~~==Example 3==~~
	~~So the final example was~~
	~~6x⁴-9x²+18 / x-3~~
	~~The trick to this question to fill in the missing terms first~~
	~~6x⁴+0x³-9x²+0x¹+18 / x-3~~
	~~Filling in these means you can apply exactly the same steps~~

	=Greatest Common Factor (GCF)=		=Greatest Common Factor (GCF)=
	This is where you look at a polynomial and find, you guest it, the greatest common factor so given		This is where you look at a polynomial and find, you guest it, the greatest common factor so given

Iwiseman at 00:50, 21 April 2026

2026-04-21T00:50:03Z

← Older revision		Revision as of 00:50, 21 April 2026
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	~~=Introduction=~~
	~~So more stuff I probably know but here because I struggle with words.~~
	~~=Terms=~~
	These are mathematical expressions with consist of a Number and a variable for example 2x. The number part is called the '''coefficient''' which is a fancy way of saying the number you need to multiply by. The variable part can be a name and raised by a power.
	~~=Polynomial=~~
	~~A polynomial is a combination of many terms. e.g 2x²+3x+24. Each term must be joined by '''addition''' or '''subtraction''' not multiplication other binary operators. There are names for terms~~
	*1 term - monomial
	*2 terms - binomial
	*3 terms - trinomial
	*More than 3 - polynomial
	~~=Examples=~~
	~~So some polynomial may not show the number or the variable part. For example~~
	~~3x² + x -5~~
	~~All this means is we have abbreviated the expression.~~
	~~3x² + 1x -5x⁰~~
	~~There now it looks proper. The last term which is just a number is known as the '''constant term''' as it never changes.~~
	~~=Degree of a term=~~
	~~The degree of a term is determined by the power of the variable part so given 2x²+3x+24~~
	~~- 2x² 2nd degree term~~
	~~- 3x 1st degree term~~
	~~- 24 constant term~~
	~~Where there are 2 variable we add them together 8x²y³ is a 5 degree term<br>~~
	~~<br>~~
	~~People refer to the whole polynomial by the highest degree to 2x²+3x+24. So this is a 2nd degree polynomial. Polynomials are arranged by degree.~~
	~~=Simplifying Polynomials=~~
	~~So like terms can be added together~~
	~~2x³+2x²+5x²+10~~
	~~Can be~~
	~~2x³+7²x+10~~
	=Long Division=		=Long Division=
	==Example 1==		==Example 1==

Iwiseman: /* Long Division */

2026-04-20T23:47:36Z

Long Division

← Older revision		Revision as of 23:47, 20 April 2026
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	2x³+7²x+10		2x³+7²x+10
	=Long Division=		=Long Division=
	==~~Examle~~ 1==		==Example 1==
	So getting more tricky below we need to follow a process. The question is		So getting more tricky below we need to follow a process. The question is
	(x² + 5x + 6) / (x + 2)		(x² + 5x + 6) / (x + 2)
Line 107:		Line 107:
	6x⁴+0x³-9x²+0x¹+18 / x-3		6x⁴+0x³-9x²+0x¹+18 / x-3
	Filling in these means you can apply exactly the same steps		Filling in these means you can apply exactly the same steps

	=Greatest Common Factor (GCF)=		=Greatest Common Factor (GCF)=
	This is where you look at a polynomial and find, you guest it, the greatest common factor so given		This is where you look at a polynomial and find, you guest it, the greatest common factor so given

Iwiseman: /* Difference in Squares */

2026-04-20T04:24:59Z

Difference in Squares

← Older revision		Revision as of 04:24, 20 April 2026
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	= (A+B)(A-B)		= (A+B)(A-B)
	= A² - AB + BA + B²		= A² - AB + BA + B²
	Now -AB + AB = 0		Now
			-AB + AB = 0
	So		So
	= A² - B²		= A² - B²

Iwiseman: /* Difference of Cubes */

2026-04-20T02:34:33Z

Difference of Cubes

← Older revision		Revision as of 02:34, 20 April 2026
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	= (5x² - 4y²)(25x⁴ + (5x²)(4y²) + 16y⁴)		= (5x² - 4y²)(25x⁴ + (5x²)(4y²) + 16y⁴)
	= (5x² - 4y³)(25x⁴ + 20x²y³ + 16y⁶)		= (5x² - 4y³)(25x⁴ + 20x²y³ + 16y⁶)
			=Zero Product Property=
			So new to me this one but obvious
			a * b = 0, then either a = 0 or b = 0
			So given
			3x² - 27 = 0
			I would do this which apparently is solving it numerically
			3x² = 27
			x² = 9
			x = 3
			But they solved it algebraically using Zero Product Property
			3(x²-9) = 0
			3(x+3)(x-3) =0
			Therefore
			x+3 = 0 and x -3 = 0
			So needed to look into this

Iwiseman: /* Difference in Squares */

2026-04-20T02:11:22Z

Difference in Squares

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 (4x² + 9y⁴)(4x² - 9y⁴)
 But the important thing to note is only difference is squares and use this approach so only right hand side
-(4x² + 9y⁴)(2x - 3y²)(2x + 3y²)
+(4x² + 9y⁴)(2x - 3y²)(2x + 3y²)<br>
 =Sum of Cubes=
 So off we go again A³ + B³ = (A + B)(A² - AB + B²)

Iwiseman: /* Sum of Cubes */

2026-04-20T01:48:01Z

Sum of Cubes

← Older revision		Revision as of 01:48, 20 April 2026
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	x³ + 8		x³ + 8
	= x³ + 2³ =(x+2)(x² - 2x + 4)		= x³ + 2³ =(x+2)(x² - 2x + 4)
			And another
			216x¹² + 343y¹⁵
			6³x¹² + 7³y¹⁵ A = 6x⁴, B = 7y⁵
			(6x⁴ + 7y⁵)(36x⁸ - (6x⁴)(7y⁵) + 49y¹⁰)
			(6x⁴ + 7y⁵)(36x⁸ - 42x⁴y⁵ + 49y¹⁰)

	=Difference of Cubes=		=Difference of Cubes=
	So off we go again again A³ - B³ = (A - B)(A² + AB + B²)		So off we go again again A³ - B³ = (A - B)(A² + AB + B²)