The logical beginning of the passage of the new topic "Polynomials" is an introductory lesson that covers the topic of multiplying a polynomial by a monomial. For starters, it’s worth giving the basic definitions that will be used in this lesson.
A monomial is a simple mathematical expression, which is a product of a constant by a variable or several variables taken in a non-negative integer without the use of the + or - signs.
A polynomial is the sum of monomials.
The main objective of this lesson is to derive a general rule for multiplying a monomial by a polynomial, as well as developing skills for applying this rule in practice. Please note that the knowledge gained from this article will be useful to each student throughout the course of studying algebra.
To study this topic, we recall the theoretical knowledge that students should receive from previous lessons. One of the necessary fragments of the theory is the properties of degrees.
Tasks for training existing skills and testing knowledge
1. It is necessary to determine the sum of the provided monomials, the difference of the monomials, the product of the monomials, the quotient of the monomials and the square of each of them.
2. Given polynomials. It is necessary to indicate the degree of each of them. The degree of a polynomial is the highest degree of the monomial that is part of the polynomial.
Having solved the training exercises, you can begin to gain new knowledge.
Character Rule
1. When multiplying two positive (+) numbers, the sign of the product does not change (+).
2. When multiplying two negative (-) numbers, the sign of the product changes to the opposite (+).
3. When multiplying a positive (+) number by a negative (-) number, the product is negative (-).
In a word, nothing complicated.
Multiplication of a monomial by a polynomial
We use the distribution law of multiplication a * (b + c) = a * b + a * c to solve the problem:
4x 3 (3x 2 - 8x + 2) = 3x 2 * 4x 3 - 8x * 4x 3 + 2 * 4x 3 = 12x 5 - 32x 4 + 8x 3.
Having knowledge of multiplying the monomial by the monomial and multiplying the monomial by a number, we calculate the value of this expression and compare the resulting expression with the above answer. It is important not to forget and correctly use the rule of placing signs.
Multiplication of a polynomial by a polynomial
Multiplying a polynomial by a polynomial in its algorithm is not very different from multiplying a monomial by a polynomial. It is simply necessary to multiply each monomial from one of the polynomials by all the monomials of which the other polynomial consists.
Self Exercise Exercise:
(5xy −3 x 2 ) * (x 2 + 3y) = 5xy * x 2 + 5xy * 3y - 3x 2 * x 2 - 3x 2 * 3y = 5x 3 y + 15xy 2 - 3x 4 - 9x 2 y.
Calculate the product of the polynomials yourself and compare the resulting expression with the answer above. It is important not to forget and correctly use the rule of placing signs.
Abbreviation Formulas
Having mastered the rules of multiplication, you can and should slightly expand your knowledge with new formulas related to this topic. It is important to understand that the formulas for abbreviated multiplication by algebra can be derived independently using the distributive property of multiplication, but in practice they are quite common, therefore it is strongly recommended to learn these formulas.
Verification required. You can independently multiply the polynomial by the polynomial and verify that after the mutual annihilation of all auxiliary terms, only the terms indicated in the formulas remain.
Geometric meaning
To simplify the understanding of the process of multiplying a monomial by a polynomial, it is worth considering the geometric meaning of this mathematical operation.
It is important here to explain to students using the example of geometric figures the distribution law of multiplication and the identical equality of both its parts.
Self Test Questions
- What is a monomial?
- What is a polynomial?
- Explain the geometric meaning of the distribution law of multiplication.
- Describe the algorithm for multiplying a monomial by a polynomial.
- What formulas of abbreviated multiplication given above did you remember?
Exercises on the topic covered
1). -3x 3 (x 2 + 3x - 4).
2). 15x 2 (xy - 4x 2 y 2 + 4).
3). -0.7m 3 n (10mn - 20m 2 - 3).
4). (2a 4 - a 3 + 0,1a) (-5a 2 ).
5). 0.5s (3s 2 d - 15c 3 d).
6). 0.5p 2 (4q - 2pq + 6p 2 ).
7). 20xy 2 (5x 2 y - 2.4y - 0.6).
eight). 4a 2 b (2a - 3ab 2 + 8b 2 ).
Test: multiplying a polynomial by a monomial
To test the knowledge gained, a special test is offered below.
| 1. Find out the work: -5x (-3x + 2x 2 - 2). a) -10x 3 - 15x 2 + 10x. b) -10x 3 + 15x 2 - 10x. c) -10x 3 - 15x 2 + 10. d) -10x 3 + 15x 2 + 10x |
| 2. Replace the sign (*) with the expression to get true equality: -2x 2 (*) = -8x 4 a 2 + 4x 2 a - 6x4. a) 4a 2 x - 2a - 3x 2 . b) 3x 2 - 2a + 4a 2 x 2 . c) 2a - 3x 2 - 4a 2 x 2 . d) -4a 2 x 2 + 3x 2 + 2a |
3. Find the product: 3a (2c - a) - 4c (c + 2a). a) 3a 2 - 4b 2. b) -3a 2 - 2ab - 4b 2. c) 3a 2 + 2ab + 4b 2. d) -3a 2 - 4b 2 |
| 4. Find the root of the given equation: 8x 2 + 3x = 0. a) x = 0. b) x = - (3/8). c) x = 0; x = (3/8). d) x = 0; x = - (3/8) |
| 5. Define the divisor of this expression: 5 4 + 5 5 + 5 6. a) 15. b) 31. at 10 o'clock. d) 55 |
Keys to the test: 1-g; 2-b; 3-b; 4-g; 5 B.
Conclusion
In this lesson, students received theoretical material with practical examples of its application on the topic "Multiplication of a monomial by a polynomial." And also learned additional material that will be useful to them in the further study of the subject. Namely, the formulas for abbreviated multiplication in algebra.
In the process of doing exercises, students developed analytical thinking.
Students had to make certain conclusions about the application of this topic in practice. Multiplication of a monomial by a polynomial applies:
- in the process of simplifying expressions;
- in the process of finding the root of the equation;
- for the purpose of proving certain identical expressions;
- in the process of solving problems in the preparation of equations.