In grade 9 students learn to multiply a binomial by a binomial. Many students have difficulty with algebra and students need a review of multiplying binomials by binomials such as ( 2x + 1)(3x - 2). Students do need to be proficient at multiplying polynomials in order to be able to check if they factored polynomials correctly.

In grade 9 students learn to multiply a binomial by a binomial. Many students have difficulty with algebra and students need a review of multiplying binomials by binomials such as ( 2x + 1)(3x - 2). Students do need to be proficient at multiplying polynomials in order to be able to check if they factored polynomials correctly.

We have been routinely adding audio signals together, and multiplying them by slowly-varying signals (used, for example, as amplitude envelopes) since Chapter 1. For a full understanding of the algebra of audio signals we must also consider the situation where two audio signals, neither of which may be assumed to change slowly, are multiplied. The key to understanding what happens is the Cosine Product Formula:

Figure 5.3: Sidebands arising from multiplying two sinusoids of frequency and : (a) with ; (b) with so that the lower sideband is reflected about the axis; (c) with , for which the amplitude of the zero-frequency sideband depends on the phases of the two sinusoids; (d) with . Parts (a) and (b) of the figure show ``general" cases where and are nonzero and different from each other. The component frequencies of the output are and . In part (b), since , we get a negative frequency component. Since cosine is an even function, we have

In the special case where , the second (difference) sideband has zero frequency. In this case phase will be significant so we rewrite the product with explicit phases, replacing by , to get:


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