1.6 Algebraic expressions and polynomials

Engineering formulas frequently combine several variable terms into a single expression that can be evaluated or simplified.

Polynomial. A finite sum of terms whose variable exponents are nonnegative integers.

A polynomial in \(x\) has the form

\[a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\]

where its exponents are nonnegative integers.

Polynomials are defined for every real input. Like terms may be combined, and products are expanded using the distributive property.

Worked Example: Combining polynomial terms

\[3x^3-9x-2x^3+4x+1=x^3-5x+1.\]

Try It. Simplify \(7t^2-3t+2t^2+5t\).
Check Your Work \(9t^2+2t\).

When multiplying polynomials, every term in one factor multiplies every term in the other. FOIL is a memory aid for two binomials, but the distributive property is the general rule. Organizing products by descending degree helps reveal like terms and makes the result easier to check.

For a channel with width \(x+3\) metres and depth \(x-1\) metres,

\[A=(x+3)(x-1)=x^2+2x-3,\]

with the physical restriction \(x>1\).

Worked Example: Expanding a channel-area expression

For width \(x+3\) metres and depth \(x-1\) metres,

\[A=(x+3)(x-1)=x^2+2x-3.\]

The physical setting also requires \(x>1\).

Try It. Expand the area of a channel with width \(x+4\) and depth \(x-2\).
Check Your Work \(x^2+2x-8\), with \(x>2\) physically.

The coefficient of the highest-degree term is the leading coefficient. A constant nonzero polynomial has degree zero. In two variables, the degree of a term is the sum of its exponents, so \(4x^2y^3\) has degree 5. Expressions containing variables in denominators, negative variable exponents, or variable square roots are algebraic expressions but are not polynomials.

Domain is determined before algebraic manipulation.

  • Polynomials accept every real input.
  • A denominator cannot be zero.
  • An even root requires a nonnegative radicand.

These rules will recur when functions are introduced.

The special-product identities provide efficient expansion for any real numbers \(A\) and \(B\):

\[ (A+B)^2=A^2+2AB+B^2,\]

\[ (A-B)^2=A^2-2AB+B^2,\]

\[ (A+B)(A-B)=A^2-B^2.\]

1.6.1 Polynomials in two variables

The expression \(5x^2y-3xy^3+7\) is a polynomial in \(x\) and \(y\). The term \(5x^2y\) has degree 3, the term \(-3xy^3\) has degree 4, and the constant has degree 0. The polynomial therefore has degree 4. Like terms must have the same variables raised to the same powers. Thus \(3x^2y\) and \(-8x^2y\) combine, but \(x^2y\) and \(xy^2\) do not.

Worked Example: Combining a two-variable flow model

Suppose two contributions to a flow model are \(4q^2t-3qt^2\) and \(-q^2t+5qt^2\). Combining like terms gives

\[(4q^2t-q^2t)+(-3qt^2+5qt^2)=3q^2t+2qt^2.\]

Try It. Simplify \((6x^2y-2xy^2)+(3x^2y+5xy^2)\).
Check Your Work \(9x^2y+3xy^2\).

1.6.2 Domains of algebraic expressions

An algebraic expression can contain roots, variable denominators, and noninteger powers. Its domain is the set of real inputs for which every operation is defined. For \(\sqrt{5-x}/(x+9)\), the square root requires \(x\le5\) and the denominator requires \(x\ne-9\). Both conditions must hold, so the domain is \((-\infty,-9)\cup(-9,5]\).

For \(\sqrt{2-x}/\sqrt{x+3}\), the numerator requires \(x\le2\). Because the square root is in the denominator, its radicand must be strictly positive, giving \(x>-3\). The combined domain is \((-3,2]\). Writing each restriction separately before taking their intersection makes complicated domains much easier to determine.

Expressions should normally be simplified only after the domain is known. A cancelled factor may remove the visible source of a restriction, but it cannot add a value that was undefined in the original expression.

Worked Example: Determining the domain of a depth formula

Consider \(D(q)=\sqrt{40-q}/(q-8)\). The square root requires \(q\le40\), and the denominator requires \(q\ne8\). Therefore,

\[\operatorname{domain}(D)=(-\infty,8)\cup(8,40].\]

Try It. Find the domain of \(\sqrt{25-x}/(x+5)\).
Check Your Work \(x\le25\) and \(x\ne-5\), so the domain is \((-\infty,-5)\cup(-5,25]\).

Applied Problem: Modelling a trapezoidal channel area. A channel has bottom width \(b\), water depth \(d\), and side slopes that add \(d/2\) metres of width on each side. Write and expand a polynomial for the cross-sectional area using average width times depth. Then evaluate it at \(b=3.0\) m and \(d=1.6\) m.

Check Your Work The top width is \(b+d\), so the average width is \([b+(b+d)]/2=b+d/2\). Thus \(A=d(b+d/2)=bd+d^2/2\). At \(b=3.0\) and \(d=1.6\), \(A=(3.0)(1.6)+(1.6)^2/2=6.08\) m².

1.6.3 Practice Problems

  1. State the degree of \(4x^5-2x+1\).
    Check Your Work 5.
  2. Simplify \(5x^2+3x-2x^2+x\).
    Check Your Work \(3x^2+4x\).
  3. Expand \((x+2)(x+5)\).
    Check Your Work \(x^2+7x+10\).
  4. Expand \((2x-3)^2\).
    Check Your Work \(4x^2-12x+9\).
  5. State the physical domain for area \(A=x(x-1)\) when both dimensions are lengths.
    Check Your Work \(x>1\).
  6. A rectangular channel has width \(d+4\) and depth \(d\). Expand its area formula.
    Check Your Work \(A=d(d+4)=d^2+4d\).
  7. Simplify the combined flow-model terms \(7q^2t-2qt^2-3q^2t+6qt^2\).
    Check Your Work \(4q^2t+4qt^2\).
  8. Find the physical domain of \(C(v)=\sqrt{60-v}/(v-10)\) when \(v\) is nonnegative.
    Check Your Work \(0\le v\le60\) with \(v\ne10\), or \([0,10)\cup(10,60]\).