"Squared Difference Formula" PPT courseware for calculation of integers

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"Squared Difference Formula" PPT courseware for calculation of integers

Question: What is factorization of polynomials?

Converting a polynomial into the product of several integers is called factoring the polynomial.

Determine which of the following deformation processes is factorization?

(1) (x-2)(x-2)=x2- 4

(2) x2- 4+3x=(x+2)(x-2)+3x

(3) 7m-7n-7=7(m-n-1)

(4) 4x2-1/9y²=(2x+1/3y)(2x-1/3y)

Question: What methods did you learn to factor?

common factor method

Factor the following expressions:

(1) ax - ay= a( x – y )

(2) 9a2 - 6ab+3a=3a(a-2b+1)

(3) 3a(a+b)-5(a+b)=(a+b)(3a-5)

(4) -4x2+8ax+2x=-2x(2x-4a-1)

Discuss

Square difference formula: (a+b)(a-b)=a²-b²

Factorization characteristics of the squared difference formula:

(1) The left side is the subtraction of the two parts (that is, the two terms have different signs)

(2) Both parts on the left can be written as the square of a certain number (or formula)

(3) The right side is the product of the sum of the two numbers and the difference between the two numbers

Formula induction

Characteristics of formulas that can be factored using the squared difference formula:

⑴ The left hand side should be a binomial (for example: 1-25b²)

⑵ Each term of the binomial (without the sign) can be written in square form.

⑶The symbols of these two items are different (for example: 25x²-4y²)

Formulas that meet the above characteristics can be factored using the squared difference formula.

Example 1. Factor the following expressions

(1)16a²- 1

(2) 4x²-m²n²

(3) (9/25)x² -(1/16)y²

(4)–9x² + 4m2

(5)x2y4-9

Factor the following expressions

1.( x + z )²- ( y + z )²

2.4(a + b)² - 25(a - c)²

3.4a³ - 4a

4.(x + y + z)² - (x – y – z )²

Use the formula method to factorize:

(1) -9x2+4y2=(2y+3x)(2y-3x)

(2) 64x2-y2z2=(8x+yz)(8x-yz)

(3) a2(a+2b)2-4(x+y)2 =(a2+2ab+2x+2y)(a2+2ab-2x-2y)

(4) (a+bx)2-1=(a+bx+1)(a+bx-1)

(5) (x-y+z)2-(2x-3y+4z)2=(3x-4y+5z)(-x+2y-3z)

summary:

1. A polynomial in the form of the difference of squares of two equations (or) two numbers can be factored using the difference of squares formula.

2. The letters a and b in the formula a² - b² = (a+b)(a-b) can be numbers, monomials or polynomials, and should be used flexibly depending on the specific situation.

3. If there are common factors in the polynomial, the common factors should be extracted first, and then further decomposed into factors.

4. Factorize thoroughly. Pay attention to the simplest form of each factor until it can no longer be decomposed.

Homework

1. If a=101, b=99, find the value of a2-b2.

2. If x=-3, find the value of 20x2-60x.

3.Is 1993-199 divisible by 200? What other integers can it be divisible by?

4. If n is an integer, prove that (2n+1)2-(2n-1)2 is a multiple of 8.

Consolidation exercises: 1. Multiple choice questions:

1) Which of the following expressions can be factored using the squared difference formula ( )

A. 4X²+y² B. 4 x- (-y)²

C. -4 X²-y³ D. - X²+ y²

2) The result of factoring -4a² +1 should be ( )

A. -(4a+1)(4a-1) B. -(2a –1)(2a –1)

C. -(2a+1)(2a+1) D. -(2a+1)(2a-1)

2. Factor the following expressions:

1) 18-2b² 2) x4 –1

Solution: 1) Original formula=2(9-b2)=2(3+b)(3-b)

2)Original formula=(x²+1)(x+1)(x-1)

3.x2-64 is factored into ( ).

(A)(x-16)(x+4); (B) (x-32)(x+32);

(C)(x+16)(x-4); (D) (x-8)(x+8).

4. 64a8-b2 is factored into ( ).

(A) (64a4-b)(a4+b); (B) (16a2-b)(4a2+b);

(C) (8a4-b)(8a4+b); (D) (8a2-b)(8a4+b).

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For more information about the PPT courseware "Operations of Square Difference Formulas and Integers", please click the "Operations of Square Differences Formula ppt Integers" ppt tag.

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Update Time: 2024-11-22

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