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Algebraic expressions. |
| Monomials | |
If you look at the following algebraic expressions you will see that they consist of different operations: Example 2.- 1) 3ax ; 2) -2xy2 ; 3) 8ab3x ; 4) 3ax - 2y ; 5) x2 + 2x - 4 In the first three expression there are no sums of the terms while in 4) and 5) there are. The first three examples are examples of monomials whereas the other two are not. We can explain why this is so: A monomial is an algebraic expression where the only operations indicated with the letters are products and powers where the exponent is a natural number. The coefficient of a monomial is the number by which the letters are multiplied in the expression. It is normally found at the beginning. If this number is 1 it is not written and this number can never be 0 otherwise the whole expression would be equal to 0. In the three examples of monomials given above the coefficients are 3, -2 and 8 respectively. The degree of a monomial is the sum of the exponents of the letters. Therefore the three monomials in the example above are: 1) degree 2, 2) degree 3, 3) degree 5 (as you know, the exponent is not written when it is 1). |
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"In all the windows in this unit the coefficients of the monomials cannot be less than -9 due to the presentation in the window." |
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"In most cases the monomials we are going to use will be the simplest type, including one coefficient and one letter only, usually x. Therefore, the corresponding exponent will indicate the degree of the monomial". For example: -2x2 ; 3x ; -5x3 ; x5 are four monomials whose degrees are 2, 1, 3 and 5 respectively. "The coefficients of a monomial may not always be whole numbers (e.g. 0.6, 1/2, -5/6 etc) but they are usually whole numbers and we shall use whole numbers in this unit".
Similar monomials are those which contain the same letters with the same exponents. Example 3: 2ax4y3 ; -3ax4y3 ; ax4y3 ; 5ax4y3 are similar monomials whereas: axy3 ; 3a2x4y3 ; 2bx4 are not similar monomials. Therefore "The only difference between two similar monomials is the coefficient". In the window in the section above if the values in the top part of the window are left unchanged and those in the lower part only are changed we get similar monomials. Note that similar monomials are of the same degree.
ADDITION AND SUBTRACTION Look at the following operations: Example 4.- 1) 5ax4y3 - 2ax4y3 = 3ax4y3
In 1) we can carry out the subtraction but in 2) we cannot carry out the addition. The first case is an example of a similar monomial whereas the second is not. Therefore: In order to add or subtract two monomials they must be similar. The result of the addition or subtraction is a similar monomial whose coefficient is the respective addition or subtraction of the corresponding coefficients. When the monomials are not similar we leave the sum in the expression and the result is a polynomial like the ones we shall be focusing on in this unit.
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- 2a) is illustrated in the window. The coefficients c1, c2 and c3 can be changed to obtain different results. - 2b) does not consist of just similar monomials. The similar monomials in the expression should be added together. (Solution: 6x3 +x ) |
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MULTIPLYING MONOMIALS In order to multiply monomials we need to remember the rules for finding the product of a power which is done when the base number is the same. For example 5x2 · 3x4 = 15x6 as: "In order to multiply powers with the same base number we leave the base number and add the exponents". Therefore, in order to multiply monomials we multiply the coefficients of each term and like base numbers raised to a power together leaving different base numbers alone. Example 5 .- Find the product of the following monomials: 4ax4y3 · x2y · 3ab2y3 . Procedure: a) Multiply the coefficients 4, 1 and 3 together. Answer: 12 b) Multiply all the terms of base a together. Answer: a2 c) Multiply all the terms of base b together. Answer: b2 d) Multiply all the terms of base x together. Answer: x6 e) Multiply all the terms of base y together. Answer: y7 Final answer: 4ax4y3 · x2y · 3ab2y3 = 12a2b2x6y7
DIVIDING MONOMIALS We cannot always divide two monomials. Look at the following examples: Example 6.- a) 4ax4y3 : 2x2y , -----------------------------b) 6x4y : ax3 In 6a) the coefficients can be divided and the letter terms of the divisor can be divided into those of the dividend, even though "a" does not appear in the divisor. The answer is a) 2ax2y2 In 6b) the division is not possible as "a" does not appear in the dividend. Division may be easier to understand if it is expressed as a fraction which we "simplify" by subtracting the exponents of like terms of the same base number: In 6b) we cannot carry out the division as we cannot simplify "a" in the denominator. "Division can also not be carried out when there is a letter in the divisor which is to a greater power than in the dividend. The answer would not be a monomial as when the exponents are subtracted we would get a negative exponent (the exponents of the letter terms should always be positive)". Example 7.- If we carry out the operation 2ax2 : (-3a3x), we get the answer - 2/3 a-2 x . The coefficient -2/3 is perfectly valid although we only usually use whole numbers as coefficients. However, a-2 is not acceptable as it is not positive.
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| Leoncio Santos Cuervo | ||
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| Spanish Ministry of Education. Year 2001 | ||
Except where otherwise noted, this work is licensed under a Creative Common License
