Remainder Theorem and Factor Theorem Practice

Identify c from x-c, then evaluate the polynomial with parentheses around a negative c.

The divisor is \(x-2\), so the Remainder Theorem gives remainder \(p(2)=9\).

Identify c from x-c, then evaluate the polynomial with parentheses around a negative c.

The divisor is \(x-3\), so the Remainder Theorem gives remainder \(p(3)=22\).

Identify c from x-c, then evaluate the polynomial with parentheses around a negative c.

The divisor is \(x--1\), so the Remainder Theorem gives remainder \(p(-1)=-6\).

Identify c from x-c, then evaluate the polynomial with parentheses around a negative c.

The divisor is \(x-1\), so the Remainder Theorem gives remainder \(p(1)=-5\).

Identify c from x-c, then evaluate the polynomial with parentheses around a negative c.

The divisor is \(x--2\), so the Remainder Theorem gives remainder \(p(-2)=-9\).

Rewrite x+d as x-(-d) before substituting; the sign in c is the key decision.

Rewrite \(x+3\) as \(x-(-3)\). Therefore the remainder is \(f(-3)=20\), not \(f(3)\).

Rewrite x+d as x-(-d) before substituting; the sign in c is the key decision.

Rewrite \(x+2\) as \(x-(-2)\). Therefore the remainder is \(f(-2)=-10\), not \(f(2)\).

Rewrite x+d as x-(-d) before substituting; the sign in c is the key decision.

Rewrite \(x+4\) as \(x-(-4)\). Therefore the remainder is \(f(-4)=-22\), not \(f(4)\).

Rewrite x+d as x-(-d) before substituting; the sign in c is the key decision.

Rewrite \(x+5\) as \(x-(-5)\). Therefore the remainder is \(f(-5)=13\), not \(f(5)\).

Rewrite x+d as x-(-d) before substituting; the sign in c is the key decision.

Rewrite \(x+1\) as \(x-(-1)\). Therefore the remainder is \(f(-1)=-5\), not \(f(1)\).

Test a candidate factor x-c by evaluating f(c); only a zero output certifies a factor.

The factor corresponds to \(c=-3\). Direct evaluation gives \(f(-3)=0\), so \(x+3\) is a factor by the Factor Theorem.

Test a candidate factor x-c by evaluating f(c); only a zero output certifies a factor.

The factor corresponds to \(c=-1\). Direct evaluation gives \(f(-1)=0\), so \(x+1\) is a factor by the Factor Theorem.

Test a candidate factor x-c by evaluating f(c); only a zero output certifies a factor.

The factor corresponds to \(c=0\). Direct evaluation gives \(f(0)=0\), so \(x\) is a factor by the Factor Theorem.

Test a candidate factor x-c by evaluating f(c); only a zero output certifies a factor.

The factor corresponds to \(c=-4\). Direct evaluation gives \(f(-4)=0\), so \(x+4\) is a factor by the Factor Theorem.

Test a candidate factor x-c by evaluating f(c); only a zero output certifies a factor.

The factor corresponds to \(c=2\). Direct evaluation gives \(f(2)=0\), so \(x-2\) is a factor by the Factor Theorem.

Convert the known factor into a zero function value, then solve the resulting linear parameter equation.

The factor gives \(p(2)=0\). The fixed terms total \(-3\), so solving \(-3+k=0\) yields \(k=3\).

Convert the known factor into a zero function value, then solve the resulting linear parameter equation.

The factor gives \(p(-3)=0\). The fixed terms total \(-2\), so solving \(-2+2k=0\) yields \(k=1\).

Convert the known factor into a zero function value, then solve the resulting linear parameter equation.

The factor gives \(p(1)=0\). The fixed terms total \(5\), so solving \(5-k=0\) yields \(k=5\).

Convert the known factor into a zero function value, then solve the resulting linear parameter equation.

The factor gives \(p(-2)=0\). The fixed terms total \(-1\), so solving \(-1+3k=0\) yields \(k=0.3333333333333333\).

Convert the known factor into a zero function value, then solve the resulting linear parameter equation.

The factor gives \(p(4)=0\). The fixed terms total \(-2\), so solving \(-2+k=0\) yields \(k=2\).

Find the row with output zero, then convert its input c into factor x-c.

The table shows \(f(-1)=0\). Therefore \(x-(-1)=x+1\) is a factor and division by it has remainder zero.

Find the row with output zero, then convert its input c into factor x-c.

The table shows \(f(3)=0\). Therefore \(x-(3)=x-3\) is a factor and division by it has remainder zero.

Find the row with output zero, then convert its input c into factor x-c.

The table shows \(f(-2)=0\). Therefore \(x-(-2)=x+2\) is a factor and division by it has remainder zero.

Find the row with output zero, then convert its input c into factor x-c.

The table shows \(f(7)=0\). Therefore \(x-(7)=x-7\) is a factor and division by it has remainder zero.

Find the row with output zero, then convert its input c into factor x-c.

The table shows \(f(0)=0\). Therefore \(x-(0)=x\) is a factor and division by it has remainder zero.

Translate f(c)=0 only into the standard zero-root-intercept-factor equivalences.

The condition \(f(5)=0\) makes \(5\) a zero/root, \((5,0)\) an x-intercept, and \(x-(5)\) a factor with zero remainder. Thus: (5,0) is an x-intercept.

Translate f(c)=0 only into the standard zero-root-intercept-factor equivalences.

The condition \(f(-2)=0\) makes \(-2\) a zero/root, \((-2,0)\) an x-intercept, and \(x-(-2)\) a factor with zero remainder. Thus: x+2 is a factor.

Translate f(c)=0 only into the standard zero-root-intercept-factor equivalences.

The condition \(f(3)=0\) makes \(3\) a zero/root, \((3,0)\) an x-intercept, and \(x-(3)\) a factor with zero remainder. Thus: Division by x-3 has remainder 0.

Translate f(c)=0 only into the standard zero-root-intercept-factor equivalences.

The condition \(f(-4)=0\) makes \(-4\) a zero/root, \((-4,0)\) an x-intercept, and \(x-(-4)\) a factor with zero remainder. Thus: -4 solves f(x)=0.

Translate f(c)=0 only into the standard zero-root-intercept-factor equivalences.

The condition \(f(1)=0\) makes \(1\) a zero/root, \((1,0)\) an x-intercept, and \(x-(1)\) a factor with zero remainder. Thus: 1 is a zero of f.

Solve each linear factor and report roots as ordered pairs on the horizontal axis.

Set each factor equal to zero. The roots are \(-4,1\), so the intercepts are \((-4,0),(1,0)\).

Solve each linear factor and report roots as ordered pairs on the horizontal axis.

Set each factor equal to zero. The roots are \(-3,2,5\), so the intercepts are \((-3,0),(2,0),(5,0)\).

Solve each linear factor and report roots as ordered pairs on the horizontal axis.

Set each factor equal to zero. The roots are \(-2,0,4\), so the intercepts are \((-2,0),(0,0),(4,0)\).

Solve each linear factor and report roots as ordered pairs on the horizontal axis.

Set each factor equal to zero. The roots are \(-1,3\), so the intercepts are \((-1,0),(3,0)\).

Solve each linear factor and report roots as ordered pairs on the horizontal axis.

Set each factor equal to zero. The roots are \(0,2,6\), so the intercepts are \((0,0),(2,0),(6,0)\).

Apply the Remainder Theorem after combining the supplied function values linearly.

Evaluate the combined polynomial at \(c\): \(2f(c)+1g(c)=2(3)+1(-2)=4\).

Apply the Remainder Theorem after combining the supplied function values linearly.

Evaluate the combined polynomial at \(c\): \(1f(c)+3g(c)=1(-4)+3(5)=11\).

Apply the Remainder Theorem after combining the supplied function values linearly.

Evaluate the combined polynomial at \(c\): \(-2f(c)+4g(c)=-2(6)+4(1)=-8\).

Apply the Remainder Theorem after combining the supplied function values linearly.

Evaluate the combined polynomial at \(c\): \(3f(c)-1g(c)=3(2)-1(-7)=13\).

Apply the Remainder Theorem after combining the supplied function values linearly.

Evaluate the combined polynomial at \(c\): \(2f(c)-2g(c)=2(-5)-2(-3)=-4\).

Evaluate the combined expression at c; a zero result certifies the matching factor.

Substitution gives \(h(2)=0\). Because the value is zero, the Factor Theorem confirms that \(x-2\) is a factor of \(h(x)\).

Evaluate the combined expression at c; a zero result certifies the matching factor.

Substitution gives \(h(-1)=0\). Because the value is zero, the Factor Theorem confirms that \(x+1\) is a factor of \(h(x)\).

Evaluate the combined expression at c; a zero result certifies the matching factor.

Substitution gives \(h(3)=0\). Because the value is zero, the Factor Theorem confirms that \(x-3\) is a factor of \(h(x)\).

Evaluate the combined expression at c; a zero result certifies the matching factor.

Substitution gives \(h(4)=0\). Because the value is zero, the Factor Theorem confirms that \(x-4\) is a factor of \(h(x)\).

Evaluate the combined expression at c; a zero result certifies the matching factor.

Substitution gives \(h(-2)=0\). Because the value is zero, the Factor Theorem confirms that \(x+2\) is a factor of \(h(x)\).

Normalize the divisor to x-c, then separate the remainder statement from the stricter factor condition.

Rewrite the divisor as \(x-c\). Here \(c=-6\); evaluate \(f(-6)\). Only a zero result establishes a factor and the intercept would be \((-6,0)\).

Normalize the divisor to x-c, then separate the remainder statement from the stricter factor condition.

Rewrite the divisor as \(x-c\). Here \(c=4\); evaluate \(f(4)\). Only a zero result establishes a factor and the intercept would be \((4,0)\).

Normalize the divisor to x-c, then separate the remainder statement from the stricter factor condition.

Rewrite the divisor as \(x-c\). Here \(c=-2\); evaluate \(f(-2)\). Only a zero result establishes a factor and the intercept would be \((-2,0)\).

Normalize the divisor to x-c, then separate the remainder statement from the stricter factor condition.

Rewrite the divisor as \(x-c\). Here \(c=0\); evaluate \(f(0)\). Only a zero result establishes a factor and the intercept would be \((0,0)\).

Normalize the divisor to x-c, then separate the remainder statement from the stricter factor condition.

Rewrite the divisor as \(x-c\). Here \(c=7\); evaluate \(f(7)\). Only a zero result establishes a factor and the intercept would be \((7,0)\).