"Multiplication Formula" Multiplication, division and factorization of integers PPT courseware 4

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"Multiplication Formula" Multiplication, division and factorization of integers PPT courseware 4

Review old knowledge——squared difference formula (a + b)(a – b)=a2 – b2

So can (a+b)(a+b) and (a-b)(a-b) also be expressed by a formula?

Explore

Calculate the following equations. What can you find?

(1) (p+1)² =(p+1)(p+1)=p² +2p+1

(2) (m+2)² =(m+2)(m+2)=m² +4m+4

(3) (p-1)² =(p-1)(p-1)=p² -2p+1

(4) (m-2)² =(m-2)(m-2)=m² - 4m+4

Formula features:

(a+b)²= a² +2ab+b²

(a-b)²= a² - 2ab+b²

1. The product is a quadratic trinomial;

2. The two terms in the product are the sum of the squares of the two numbers;

3. The other term is twice the product of the two numbers, and has the same sign as the middle of the multiplication.

4. The letters a and b in the formula can represent numbers, monomials and polynomials.

Does the following equation hold true? Explain why.

(1) (4a+1)²=(1−4a)²;

(2) (4a−1)²=(4a+1)²;

(3) (4a−1)(1−4a)=(4a−1)(4a−1)=(4a−1)²;

(4) (4a−1)(1−4a)=(4a−1)(4a+1).

Reasons: (1) From the commutative law of addition -4a+l=l−4a.

(2) ∵ 4a−1＝-(4a+1),

∴(-4a−1)²＝[-(4a+1)]²＝(4a+1)².

(3) ∵ (1−4a)＝−(-1+4a)＝-(4a−1),

That is (1−4a)=-(4a−1)

∴ (4a-1)(1-4a)=(4a-1)·[-(4a-1)]

=-(4a-1)(4a-1)=-(4a-1)².

(4) The right hand side should be:-(4a−1)(4a+1).

summary

Note that the formula for perfect squares is different from the formula for square variances:

Different forms.

The results are different:

The result of the perfect square formula is three terms, namely (a±b)²=a²±2ab+b²;

The result of the squared difference formula is two terms, (a+b)(a−b)=a²−b².

In the process of solving the problem, it is necessary to accurately determine a and b, compare both sides of the prototype of the formula, and ensure that no terms are lost, no symbols are mistaken, and 2 ab is multiplied by 2 at least; when the first (second) number is the square of the product, be careful to add Parentheses are the key to using the perfect square formula for polynomial multiplication

Sometimes it is necessary to deform it so that the deformed formula meets the conditions for applying the perfect square formula, that is, "the square of the sum (or difference) of two numbers", and then apply the formula to calculate.

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Update Time: 2024-10-21

This template belongs to Mathematics courseware People's Education Press Eighth Grade Mathematics Volume 1 industry PPT template

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